Speedup in the Traveling Repairman Problem with Constrained Time Windows

Sp eedup in the T ra v eling Repairman Problem with Constrained Time Windo ws Greg N. F rederic kson ∗ Barry Wittman † No v em b er 21, 2018 Abstract A bicriteria a ppro ximation algorithm is pres en ted for the unro oted traveling repairman pro b- lem, realizing increased pro ﬁt in return for incre ased speedup of repairman motio n. The algo - rithm generalizes previous results from the case in which all time windows ar e the same length to the ca se in whic h their leng ths c a n range be tw een l and 2. This ana ly sis ca n ex tend to any range of time window lengths, following our earlie r tec hniques [1 1]. This rela tionship be tw een repairman pro ﬁt and sp eedup is a pplicable over a rang e of v alues that is dep enden t o n the c o st o f putting th e input in an esp ecially desirable form, in volving wha t are ca lled “trimmed windows.” F or time windows with leng ths b etw een 1 and 2, the ra nge o f v alues for sp eedup s for which our analysis holds is 1 ≤ s ≤ 6. In this range , w e establish an a ppro ximation ratio that is constant for a n y s p eciﬁc v alue of s . Key wor ds: Approximation algo rithms, time windows, trav eling r epairman, TSP 1 In tro duction In this pap er we present an appr o ximatio n algorithm for a practical time-sensitiv e r outing problem, the un rooted tr a v eling repairman problem w ith time windows. The inpu t to this pr oblem is a sp eed at whic h a repairman can tra ve l and a list of servic e r e quests . E ach service request is lo cat ed at a no de in a w eigh ted metric graph, whose edges giv e the trav el distance b et w een no des. Each service request also has a sp eciﬁc time window d u ring which it is v alid for ser v ice. The goal of the problem is to plan a route called a servic e run that, starting at an y service r equest at an y time, visits as man y ser v ice requests as p ossible during th eir resp ectiv e time windo ws. Because the problem is NP-hard, our only hop e for an eﬃcient approac h seems to be an ap- pro ximation algo rithm. In the real w orld, a r epairman may ha v e some ﬂexibilit y in c ho osing sp eed. As a consequence, our earlier appro ximation algorithms [12] and this pap er are parameterized b y sp eedup s , so that w e can c haracterize how m uc h closer to optimal the repairman can d o if he or she tra ve ls a factor of s faster than a hypothetical repairman tra v eling alo ng an optimal route at the baseline sp eed. This type of appro ximation based on resource augmen tation is well kno wn in the scheduling comm unit y as shown b y Bansal et al. [3], Kalya nasun daram and Pru h s [13], and Phillips et al. [16]. The algorithms in th is pap er bu ild on our earlier wo rk [10, 11], in w hic h we in tro duced the ﬁr s t p olynomial-time alg orithms that giv e constan t approxima tions to the tra v eling repairman problem ∗ Dept. of Computer S ciences, Purdu e Universit y , W est Lafay ette, IN 47907. gnf @cs.purdue.edu † Dept. of Computer S cience, Elizab eth town College, Elizab eth town, P A 17022. wit tmanb@etown.edu 1 when all the time w indo ws are the same length. As a count erp oint to the repairman pr oblem, w e also in tro duced the sp eeding d eliv eryman problem in [10, 11], w ith an alternativ e optimiza tion paradigm, namely sp eedup. The input to th e sp eeding deliv eryman problem is the same as the input to the tra v eling r epairman problem, but the goal is to ﬁnd the minimum sp eed necessary to visit al l service requests du ring their time windo ws and thus collect all proﬁt. In [11] we also ga v e co nstant-fa ctor p olynomial-t ime appro ximation algorithms for b oth p r oblems w h en th e time windo ws ha v e lengths in some ﬁxed range. In b oth the repairman and deliv eryman problems, our algorithms [10, 11] r ely on trimming windo ws so that th e resulting time windows are pairwise either iden tical or n on-o v erlapping . W e trim time windo ws b y rep ea tedly making divisions in time after a ﬁxed amoun t of time has passed, starting at a sp eciﬁed time. W e deﬁne a p erio d to b e the time interv al that starts at a particular division and con tin ues u p to the next division. When time windo ws are unit length, we c ho ose a p erio d length of .5 time u nits. Bec ause we d eﬁne p eriods so th at no window starts on a p erio d b oundary , eac h time windo w will completely ov erlap exact ly one p erio d and partially o ve rlap its t w o neigh b oring p erio ds. T rimming th en remo v es those parts of eac h wind o w that fall outside of the completely o v erlapp ed p erio d. In th is simpler case where time windo ws all ha ve the same length, the p enalt y for trimming the repairman is a redu ctio n b y a factor of 1 / 3 in the n umb er of requests serviced, and the p enalt y for the d eliv eryman is a n increase by a facto r of 4 in the sp eed needed to service all requests. In [12], w e sho we d that, for unit time wind o ws, a sp ectrum of p erformance is p ossible b et w een these t w o extremes. F or some sp eedup s greater than 1 but less than 4, w e sho w ed ho w to ac hieve an increase in th e num b er of serviced requests, prop ortional in some sense to s . The appro ximation is also a function of graph prop ert y γ , w here γ = 1 for a tree and γ is no more than 2 + ǫ for a metric graph, for an y constant ǫ > 0. A more complete explanation of γ is giv en in S ect. 2. In this p aper, w e extend our algorithms and analysis to the more challe nging case in wh ic h time windo ws ha ve lengths in some ﬁxed range, sp eciﬁca lly b et wee n 1 and 2. W e presen t an algorithm that ﬁnds approximat ions parameterized b y sp eedu p s and pr operty γ . T o pro v e these app ro xi- mation b oun ds, our analysis establishes and tak es adv an tage of the existence of an ensem ble of runs that mo v e bac kw ard and forward along the path of an (unkno wn) optimal run , similar to our w ork in [12]. These runs are analyzed based on sev eral d iﬀeren t starting p oin ts for trimming. T o handle windo ws of diﬀeren t lengths (i.e ., b et wee n 1 and 2), we orc hestrate sev eral complemen tary trimming sc hemes, run our app ro ximation algorithm on eac h com bination, and c ho ose the b est result. On the surface, th e appr oac h w e u se to orc hestrate trimming schemes is similar to our app roac h in [11], whic h extended our ea rlier approximat ion algorithms from [10] to ac h iev e a constant ap- pro ximation on time windo ws whose lengths w ere b et w een 1 and 2. How ev er, the similarit y of the algorithms b elies the fundamen tal diﬀerence in the analysis, whose complexit y increases b y at least an order of magnitude in the pro cess of uniting sp eedup w ith non-u n iform time wind o ws. The k ey to our algorithm remains using a diﬀeren t p erio d length for eac h tr im m ing sc heme, with eac h subsequent s c heme using a progressiv ely longer p erio d length. In tuitive ly , by selecting the most proﬁtable r u n found in an y sc heme, the algorithm adapts to diﬀeren t distributions of windo w sizes. If most of the windows are short, a sc heme of trimming to sh orter lengths will b e eﬀectiv e. If most of the wind o ws are long, a sc heme of trimmin g to longer lengths w ill b e eﬀec tiv e. Because the output of eac h trimm in g s cheme is a set of trimmed wind o ws of equal length, our sp eedup algorithms from [12] can then b e app lie d directly . As with the case of no sp eedup, w e b ound the appro ximation guaran tee of our algorithm by accoun ting for a v ariet y of distributions of w indo ws, but the to ol needed to b ound eac h distribu tio n is n o w a considerably ric her set of h yp othetical runs. 2 The ma jor con tributions of th is pap er are tw o additional tec hniqu es n ee ded to extend the analysis f or sp eedup on unit-time wind o ws to windo ws with non-uniform length. T he ﬁrst tec hnique is a signiﬁcant ly more complex design of ensem bles to ac hiev e go od co v erage, usin g a greater v ariet y of runs, some of whic h ha v e longer rep eating patt erns . Once w e select an appropriate ensemble, w e use a symb oli c description of the co v erage of the runs in the ensem ble to demonstrate go o d co v erage for all sp eedups in the range of sp eedups under consid erati on. The sec ond tec hn ique is an approac h for d esigning and coord inating tog ether the diﬀeren t b oun d s of approximati on as a function of sp eedup for diﬀeren t win do w lengths. Usin g a v eraging argumen ts, we will sh o w that an y con v ex com bination of th e approximat ion guarant ees for eac h trimmin g sc heme is a lo w er b ound on the proﬁt of the b est run pro duced b y our algorithm. F or eac h range of sp eedups in qu esti on, w e d ete rmin e the b est choic es of weigh tings for a conv ex combination of th e app ro ximation b ounds w e h a v e foun d. By using the b est conv ex com bin at ion of b ound s from eac h sc heme, w e gu arantee a go o d b ound of appro ximation. T he d eta ils of these tec hniqu es are giv en for the case when windo w size is b et w een 1 and 2, b u t other ranges of window size can b e accommo dated in a similar w a y . As a result, w e can still p r odu ce p olynomial-ti me appro ximation algorithms with constant- factor app ro ximations for a giv en s o v er a signiﬁ cant sp eedup range. Our pro cess of combining together diﬀerent appro ximation b ound s, as a f unction of the sp eedup s , give s a ﬁ nal result in T able 1 that is more inv olv ed than our resu lts in [12]. The ratio has more piecewise ranges an d its in ve rse is primarily n onlinear, ev en though the in v erse of the ratio in eac h range is fairly close to a linear function. Note that app ro ximatio n ratios are typically deﬁn ed to b e at least 1, and so these appro ximation ratios will giv e the recipro cal of the fraction of pr oﬁt collecte d at a giv en sp eedup. F or ease of present ation, most of th e analysis in th is pap er will instead b e in terms of the fraction of p roﬁt co llected. Upp er Bound on Appro ximation R atio Sp eedup 219 γ / (26 s + 26) γ (28 s 2 + 24 s + 12) / (5 s 3 + 6 s 2 ) γ ( − 4 s 3 + 40 s 2 − 12 s + 8) / ( s 4 − 2 s 3 + 11 s 2 ) γ (68 s 3 − 172 s 2 − 140 s − 92) / (11 s 4 − 21 s 3 − 50 s 2 ) γ (292 s 3 − 1636 s 2 + 2672 s − 1472) / (39 s 4 − 183 s 3 + 180 s 2 ) γ (12 s 2 + 8 s + 16) / ( s 3 + 6 s 2 ) γ ( − s + 16) / ( s + 4) γ (3 s − 26) / ( s − 14) 1 ≤ s ≤ 2 2 ≤ s ≤ 7 3 7 3 ≤ s ≤ 17 7 17 7 ≤ s ≤ 5 2 5 2 ≤ s ≤ 3 3 ≤ s ≤ 4 4 ≤ s ≤ 5 5 ≤ s ≤ 6 T able 1: Appro ximation ratios for sp eedup s when time windo w lengths are b et wee n 1 and 2. Our results are recen t dev elopment s in time-sensitive routing problems, whic h ha ve r ece ive d a lot of atten tion from the algorithms communit y in the last d ec ade. As with our particular p roblem, these p roblems t ypically iden tify the locations to b e visited and the cost of tr a v eling b et w een them as the no des and edges, resp ect ivel y , of a w eigh ted graph. F or example, the orien teering pr oblem considered by Ar kin et al . [1], Ba nsal et al. [2], Blum et al. [5], Cheku r i et al. [7], an d Chen and Har-P eled [9] s eeks to ﬁnd a path that visits as many no des as p ossible b efore a global time d ea dline. The deadline trav eling salesman problem whic h w as also consid er ed by Bansal et al. [2 ] generalizes this problem further b y allo wing eac h location to hav e its o wn deadline. Our tra v eling repairman 3 problem can b e view ed as a fu r ther generaliz ation from a deadline to a time win do w. A great deal of wo rk b y Bansal et al. [2], Bar-Y e huda et al. [4], Chekuri and K umar [8], Karuno et al. [14], Tsitsiklis [17], and the authors [11] has b een done on the tra v eling rep airm an problem, although m uc h of the preceding literat ur e, including that from Ba nsal et al. [2] and Ba r-Y eh ud a et al. [4], considers the ro oted version of the problem, in whic h the repairman starts at a sp ec iﬁc lo ca tion at a sp eciﬁc time. F or general time windo ws in the ro oted problem, an O (log 2 n )-appro ximation is giv en by Bansal et al. [2]. An O (log L )-appro ximation is giv en b y Chekuri and Korula [6], for the case that all time windo w start and end times are int egers, w here L is th e length of the longest time wind o w. In con trast, a constan t appro ximation is giv en b y Chekuri and Ku mar [8], b ut only when there are a constan t n um b er of diﬀeren t time windo ws. Our earlier w ork [11] an d w ork by Chekur i and K orula [6] give O (log D )-approximat ions to the un rooted problem with general time windows, w here D is the ratio of the length of largest time window to th e length of the smallest. P olylogarithmic appro ximation algorithms for the dir ec ted trav eling salesman p roblem with time windows ha v e also b een giv en b y Cheku ri et al. [7] and Nagara jan and Ravi [15]. 2 T rimming T ime Windo ws and the Asso ciated Loss In our earlier w ork [11, 12 ], w e trimm ed time windo ws of unit length by ﬁrst making d ivisio ns in time ev ery . 5 time units, starting at time 0. W e generalize the pro cess f or time wind o ws with diﬀerent lengths b y instead making divisions ev ery α time u nits, wh ere α = . 5, . 75 , or 1 for the range of time windo w lengths [1 , 2). Deﬁne a p erio d to b e the time interv al that starts at a particular division and con tin ues up to the next division. In the case of u nit time wind o ws, eac h time window w ill completely o v erlap exactly one p eriod and partially ov erlap its t w o neigh b oring p eriods, b ecause w e allo w no window to start on a p erio d b oundary . In th e case of longer time w indo ws, there will b e diﬀeren t patterns of o v erlapping. If a windo w completely o verla ps with only one p eriod , trimming will remo v e those parts of eac h wind o w that fall outsid e of the completely o v erlapp ed p erio d. If a w indo w completely o ve rlaps with more than one p erio d, one trimming s cheme will remo v e all those parts of eac h suc h window that fall outside of the ﬁrst completely ov erlapp ed p erio d. An ot her separate trimmin g sc heme will remov e all those p arts of ea c h suc h windo w that fall outside of the second complete ly o v erlapp ed p erio d. F or long p eriod sizes, some time win do ws ma y not co mpletely o verla p any full p eriod s and will v anish in the pro cess of trimming. In eac h case, b ecause p erio ds do not o v erlap and a time windo w is trimmed to at most one p erio d, trimmed time w in do ws will b e pairwise either identica l or non-ov erlapping. Our repairman algorithm in [11] iden tiﬁes a v ariet y of go od paths in side eac h separate p erio d and then us es d ynamic p rogramming to select and paste these paths together into a v ariet y of longer go o d paths and ultimately a go o d service ru n for the w hole problem. T o describ e our results for b oth trees and general metric graphs, we use the graph prop ert y γ , where γ = 1 for a tree, derived in ou r earlier pap er [11], and γ ≤ 2 + ǫ for a metric graph, deriv ed b y Chekur i et al. [7 ], for any constan t ǫ > 0 . T o describ e the runnin g time for these repairman algorithms we use Γ( n ), where Γ( n ) is O ( n 4 ) f or a tree and O ( n O (1 / ǫ 2 ) ) for a metric graph. The v alue of γ and runnin g time of Γ( n ) are dep endent on the appro ximation b ounds for ﬁnd ing maximum proﬁt paths within a sp eciﬁc p erio d on a sp eciﬁc class of graph. Although the a v aila ble results only giv e γ v alues and Γ( n ) running times for trees and metric graphs, other classes of graphs, suc h as outerplanar or Euclidean graphs, ma y hav e intermediate v alues of γ and Γ( n ). In [11] w e sho wed that, for unit time win do ws with no sp eedup, the reduction du e to trimming still allo ws us to visit at least 1 / (3 γ ) of the pre-trimming optimal and , with a sp eedup of 4, w e 4 can visit at least 1 /γ of the pr e-t rimm ing optimal. In [12] we ﬁlled in the gap b et wee n these t w o extremes with an 6 γ / ( s + 1)-approximati on f or sp eedup in the r ange 1 ≤ s ≤ 2 and a 4 γ /s - appro ximation f or sp eedup in the range 2 ≤ s ≤ 4. W e contin ue to demonstrate the ﬂexibility of trimming in th e realm of sp eedup by extendin g these results to time w indo ws with diﬀerent lengths. 3 The Ensem ble Ap proac h for Analyzing P erformance Giv en an instance of the repairman pr oblem on unit time windows, our previous w ork [12] pr esented algorithms for rational sp eedup s = q /r in the range 1 ≤ s ≤ 4 that tak e O (min { r , m } Γ( n )) time, where m is the n umb er of distinct p erio ds. Since the approximati on fun ction is smo oth and con tin uous, those algorithms w ork f or an y real sp eedup s , in the same r ange, in O ( m Γ( n )) time. The analysis of these algorithms uses a num b er of d iﬀeren t service runs on trimmed time windo ws that are based on mo ving bac kw ard a nd forw ard al ong an optimal tour R ∗ . W e rely on a v eraging ov er a suitable ensem ble of runs to establish that some run R on trimmed time windo ws do es well. Because w e will build on this tec hniqu e and, indeed, use some of th e same runs from our earlier wo rk [12 ] in ensembles for v ariable length win do ws, we will r eview our n ota tion for describing these runs. W e deﬁne unit sp eed to b e some reference sp eed. T ra ve ling with s = 1 is trav eli ng at unit sp eed. Our results will hold whenever u nit sp eed is n o faster than the slo w est sp eed at whic h an optimal service run is able to visit all lo cati ons during their time wind ows. Intuitiv ely , this restriction means that w e are focusin g on those cases when unit s peed is lo w enough that sp eeding up our ser v ice r uns will actually giv e some b eneﬁt. Let R ∗ b e an optimal service run at unit sp eed originally starting at time 0. In o ur analysis we use the term r acing to describ e mo v ement , forwards and bac kw ards, along R ∗ at a sp eedu p of s times unit sp eed. Not e that our analysis o f run co v erage is describ ed on a p erio d length of . 5 even though, in Sect. 8, we will apply this analysis to our algorithm, whic h uses three d iﬀeren t p erio d lengths. Deﬁne service r un A as follo ws. Start ru n A at time t = 0 at the location that R ∗ has at time t = − 0 . 5. Then run A follo ws a pattern of racing forward along R ∗ for 1 p erio d, r aci ng bac kward along R ∗ for 1 − 1 /s p erio ds, and then racing forward along R ∗ for 1 /s p erio ds. Note that the pattern of mo v emen t for r un A rep eats ev ery 2 p erio ds. Considering the pr ob lem in whic h windo ws h a v e length b etw een 1 and 2, let λ b e an upp er b ound on the num b er of p eriod s fully conta ined in a win d o w. Deﬁne A R , the “ r everse ” of A , as follo ws. When run on a set of requests whose wind o ws eac h fully con tain at most λ p erio ds, run A R starts at time t = 0 at the lo ca tion that R ∗ has at time t = λ/ 2. Then r un A R follo ws a rep eating pattern of racing forwa rd along R ∗ for 1 / s p erio ds, racing bac kwa rd along R ∗ for 1 − 1 /s p erio ds, and then r ac ing forw ard along R ∗ for 1 p erio d. Figure 1 shows examples of runs A and A R with a sp eedup of 2 when λ = 1. F or th e pu rp oses of analyzing our run A , num b er the p erio ds 0, 1, 2, and so on by the in teger m ultiples of . 5 that giv e their starting times. Ru n A rep eats every 2 p erio ds, and its co v erage v aries dep ending on whether the p erio d n um b er is eve n or o dd. T o b ala nce this asymmetry w e deﬁn e ~ A and ~ A R , shifted v ersions of A and A R , resp ec tive ly . Run ~ A follo ws the same patte rn as A but starts the p att ern at time . 5 at the lo catio n R ∗ has at time 0. Run ~ A R follo ws the same p at tern as A R but starts the pattern at time . 5 at the location R ∗ has at time λ/ 2 + . 5. Our analysis w ill require sev eral versions of A that ha v e d iﬀerent starting p oin ts. T o simplify notation, for any giv en rational sp eedup s = q /r , le t a hop b e the amoun t of distance tra v eled in 1 / (2 r ) time at unit sp eed. Let A ∆ b e th e run A mo v ed forw ard ∆ hops and let run A R ∆ b e the 5 Run A R Optimal Run R ∗ Run A S 0 S 1 S 0 S 0 S 1 S 0 S 0 S 2 S 1 S 2 S 2 S 1 0 . 5 1 1 . 5 − . 5 2 0 . 5 1 1 . 5 2 0 . 5 1 1 . 5 Figure 1: Examples of runs A and A R with a sp eedup of 2 w hen λ = 1 based on an optimal run . Times are lab eled on the optimal run as w ell as runs A and A R . Segmen ts of eac h r un are also designated S 0 , S 1 , and S 2 dep ending on whic h su bset of ru ns they add co v erage to. This su bset naming scheme will b e f ully explained in S ect. 4. run A R mo v ed b ac kw ard ∆ hops. Run A ∆ follo ws the same pattern of mo ve ment as run A but starts at time t = 0 at the location that R ∗ has at time t = − 0 . 5 + ∆ / (2 r ). Run A R ∆ , the rev erse of A ∆ , follo ws the same pattern of mo v emen t as A R but starts at t = 0 at the lo cation that R ∗ has at t = λ/ 2 − ∆ / (2 r ). These reve rsed and s hifted v ersions of A were required to establish the p erformance of our alg orithms in [12] on unit length windo ws. Although run A with its 2-p erio d rep eating pattern is suﬃcien t for those cases, we will in tro duce additional runs whic h rep eat after 3 or 4 p erio ds in order to handle win do ws of longer length. If a service request p is serviced b y a ru n R du r ing the p eriod that the time window of p h as b een trimmed into, w e sa y that R c overs p . Let S b e a sub s et of service requests. Deﬁne the c over a ge of S by a ru n R , wr itt en cov er R ( S ), to b e the n umber of requests in S co v ered by R d ivided b y the n umb er of requests in S . Deﬁne the co v erage of S b y a set U of runs, written cov er U ( S ), to b e the a v erage of cov er R ( S ) for ev ery run R ∈ U . W e will still rely on our Av erage Co ve rage Prop osition from our earlier work [12]: Prop osition 3.1 (Average Co v erage) L et { S 1 , S 2 , S 3 , ...S a } b e a c ol le ction of sets of servic e r e quests such that S i S i gives al l the servic e r e quests servic e d by R ∗ on untrimme d windows. L et U b e a set of servi c e runs. If min i { cov er U ( S i ) } = µ , then ther e is at le ast one servic e run ˆ R ∈ U such that pr o ﬁt ( ˆ R ) ≥ µ · pr o ﬁt ( R ∗ ) . The Ave rage Co v erage Prop osition formalizes th e follo wing in tuition. Let a group of service runs ac hieve some co v erage ov er a set of requests. Let us also say that we ha v e divided those requests into man y diﬀerent su bsets, some of whic h may ov erlap, but the union of all the subs ets is the original set of requests. If w e tak e the subset of requests with the w orst a v erage co v erage, some service run in th e group co ve rs a fraction of tot al requests no smaller than that wo rst a v erage co v erage. Oth erwise, the a v erage co verage of all subsets w ould b e w orse than the co v erage of the w orst co v ered subset, which is a contradicti on. Giv en a wa y of dividing requ ests in to s u bsets, we wish to pro ve that some set of service runs ac hiev es some lo w er b ound on a v erage co v erage. The follo wing sectio n will describ e the algo rithm w e will use to ﬁnd a service ru n and the analytical tec hniques we will use to establish a lo w er b oun d on its p erf ormance. This analysis will dep end on carefully sho wing an a v erage co v erage for v arious subsets of requests deﬁned with resp ect to p erio ds indu ced by trimm in g. 6 4 Algorithm for Windo ws with Lengths b et w ee n 1 and 2 In [11], we describ e algorithms for s = 1 that ac hiev e constant app ro ximations when windo w sizes are not necessarily u niform but are close to b eing uniform. W e extend that approac h f or our sp eedup problem, but sp eciﬁcally for windo ws whose lengths diﬀer by at m ost a factor of 2. F rom [12], we giv e an algorithm called S PEEDUP that, for un it wind o ws, ﬁnds a run of app ro ximately optimal pr oﬁ t at sp eedup s . Our approac h is to mo dify SPEEDUP and run it with three diﬀeren t sizes of p eriod α : .5, .75, and 1. F or eac h p erio d size, w e will consider m ultiple starting p oints for a set of p erio ds, eac h spaced .25 apart. W e mo dify S P EEDUP appropriately s o that n o windo w starts at the b eginnin g of a p erio d, for p er io ds of size .5, .75, or 1. Th is mo diﬁed algorithm is called SPEEDUPW12 and is sp eciﬁed b elo w. In this algorithm, sets of p erio ds whose α is .5, .75, or 1 ca n ha ve 2, 3, or 4 unique starting p ositions, resp ectiv ely . Dep ending on a giv en p erio d size α a nd starting p oint, a w indo w will partially ﬁll 2 p erio ds and fully ﬁ ll 0, 1, 2, or 3 p erio ds b et w een the 2 partial p erio ds. F or ℓ = 1 , 2 , 3 and a sp eciﬁed v al ue of α , let W ℓ b e th e set of windows that completely ﬁll exactl y ℓ subinterv als and partially o v erlap with tw o m ore of them. SPEEDUPW12 PHASE 1: Set α to .5 and iden tify windo ws for sets W 1 , W 2 , and W 3 . F or i from 0 to 1, Set the starting p oin t for the p erio ds to i/ 4. F or j from 1 to 2, F or k from 1 to 3, T rim eac h windo w in W 1 to its 1 st subinterv al. T rim eac h windo w in W 2 to its j th subinterv al. T rim eac h windo w in W 3 to its k th subinterv al. Run SPEE DUP and retain the b est result so far. PHASE 2: Reset α to .75 and then identi fy windows for W 1 and W 2 . F or i from 0 to 2, Set the starting p oin t for the p erio ds to i/ 4. F or j from 1 to 2, T rim eac h windo w in W 1 to its 1 st subinterv al. T rim eac h windo w in W 2 to its j th subinterv al. Run SPEE DUP and retain the b est result so far. PHASE 3: Reset α to 1 an d then identify windo ws for W 1 . F or i from 0 to 3, Set the starting p oin t for the p erio ds to i/ 4. T rim eac h windo w in W 1 to its 1 st subinterv al. Run SPEE DUP and retain the b est result so far. When trimm ing, we c ho ose from sev eral c hoices of whic h single full subint erv al to k eep for eac h windo w. F or example, for p erio ds of length .5 and for wind o ws in W 3 whic h w ould h a v e th ree full subinterv als, the c hoices for trimming will b e to trim the windo w d o wn to either th e ﬁrst, second, or third full subin terv al. Com bining these c hoices with the t w o choic es asso ciated with windo ws in 7 W 2 and the single c hoice in windows in W 1 yields 6 trimmings. The p erformance for sp eedu p for windo ws in W 1 in the range 1 ≤ s ≤ 4 is the same as the u n it time windo w resu lts give n by our work in [12]. In Sect. 5, we giv e the p erformance for sp eedup for windo ws in W 2 in the range 1 ≤ s ≤ 5. In S ect. 7, w e giv e the p erformance for sp eedup for windo ws in W 3 in the range 1 ≤ s ≤ 6. Note that the SPEEDUP subroutine w orks for any real n umb er 1 ≤ s ≤ 6; ho we ve r, our analysis will assume that s is a rational n um b er such that s = q /r . In the ca se that s is irrational, our analysis holds in the limit b ecause the functions w e ﬁnd that b ound p erformance in terms of s are piecewise smo oth and con tin uous. I t is worth rep eating that our analysis uses only a p erio d size of .5 but can still b ound the p erformance of our algorithm with its thr ee diﬀeren t p erio d lengths b y usin g careful accoun ting of subset co v erage . When d ea ling with win do ws of unit length in a p revious pap er [12], w e d eﬁned a partitio n of requests into three sets, based on whic h p erio d a request wa s tr im m ed in to v ersus whic h p erio d an optimal ru n R ∗ serviced the r equest in. S et T consists of r equests serviced by R ∗ in the same p erio d, set E consists of requests serviced b y R ∗ in th e preceding p erio d , and set L consists of requests serviced b y R ∗ in the follo wing p eriod . F or windows of length b etw een 1 and 2, w e need to extend this approac h. W e will use a sup erscripted S to designate that a requ est that was serviced b y R ∗ in either the ﬁrst, second, third, fourth, or ﬁ fth p erio ds with whic h a request o v erlaps. F or requests in W 1 , the sets L , T , and E will b e renamed S 0 , S 1 , and S 2 , resp ectiv ely . F or requ ests in W 2 and W 3 , w e will go fur th er and use d esignat ions S 0 through S 3 and S 0 through S 4 , resp ectiv ely . Set S 3 consists of requests serviced b y R ∗ t w o p erio ds b efore the p erio d in to whic h those requests w ere trimmed, and S 4 consists of those requ ests serviced three p eriod s ea rlier. In the same earlie r w ork [12], we further partitioned L , T , and E in to L j , T j , and E j for j = 1 , 2 , . . . , r . I n a s im ilar w a y , we will partition sets S 0 , S 1 , S 2 , S 3 , and S 4 in to r equal-length divisions, subs et s S 0 j , S 1 j , S 2 j , S 3 j , and S 4 j , for an y given j , j = 1 , 2 , . . . , r . Let [ w , w + k ) b e an y time wind o w, where 1 ≤ k ≤ 2. Let ω b e the smallest integer m ultiple of 1 / (2 r ) that is greate r than w . W e designate subin terv als [ w, ω ), [ ω , ω + 1 / (2 r )), [ ω + 1 / (2 r ) , ω + 2 / (2 r )), . . . , [ ω + (4 r − 1) / (2 r ) , w + 2) by w 0 , w 1 , w 2 , . . . , w 4 r . F or windo ws of length betw een 1 and 2 with a giv en c hoice of p eriod starting times, all windows f all in to set W 1 , W 2 , or W 3 . In our analysis, there is alw a ys an implied factor of γ that acco unts for the diﬀerence b et ween the appro ximation on a tree and on a metric graph. W e no w d eﬁne a pro cedure called CREA TE-T ABLE- λ that describ es the pro cess of determining co v erage for a particular sp eedup s for a particular run mo v ed forw ard ∆ hops. This pro cedure is a generalizatio n of our CREA TE-T ABLE pro cedure from [12] to λ ≥ 1. (Recall that λ is an upp er b ound on th e num b er of p erio ds fully con tained in a window.) Note that CREA TE-T ABLE- λ is n ot an algorithm that is run in the pro cess of ﬁnding an app ro ximation to a repairman problem with sp eedup. Rather, it p ro vides a template that ca n b e u sed to pro duce the tables used in analyzing the per f ormance of s u c h appr oximati ons. So that the treatmen t here is self-con tained, we rep eat m uch of our discussion of table construction from [12], mo difying it as necessary so th at it ca n also handle the additional t yp es of runs th at w e will in tro duce. Before CREA TE-T ABLE- λ can b e completely deﬁn ed , it is n ece ssary to explain the pattern of co v erage ge nerated b y a run. F or the kind of runs w e h a v e seen so far, t yp e A runs, this p att ern tak es one of t w o forms . Let s b e a r ati onal num b er s u c h that s = q /r . T yp e A runs rep eat ev ery t w o p erio ds and th us can b e repr esen ted with a pattern of cov erag e that u s es a 1 to signify a subset co v ered ev ery p eriod and a 1/2 to signify a subset co v ered ev ery other p erio d. Observe the mo v emen t of t yp e A r uns, noting that, durin g its ﬁrst p erio d, suc h a r un mov es forw ard the same distance that an optimal r un mov es during q su b in terv als. During its second p erio d of time, it mov es backw ard th e same distance than an optimal run mo v es du ring q − r 8 subinterv als and then forw ard the same distance th an an optimal run mo v es du ring r subinte rv als. Then, the pattern r epeats. When s < 2, ru n A , during the ﬁrst p erio d in its pattern, co v ers q successiv e su bsets as it mo v es forw ard, wh ile in its second p erio d co v ers r su bsets as it mo v es forw ard. Note that those subsets co v ered as A mo v es bac kwards ad d nothing add itio nal to the co v erage. Thus, this pattern of co v erage is represen ted as r rep etitions of 1 and q − r rep etitio ns of 1/2. When s ≥ 2, run A , d uring th e ﬁrst p erio d in its pattern, co v ers q s ubsets subin terv als, while in its second p erio d co v ers q − r su bsets bac kw ards but no new subsets f orw ard. Th is pattern of co v erage is repr esen ted as q − r r epetitions of 1 and r repetitions of 1/2. CREA TE-T ABLE- λ (hops ∆ ) Let the ﬁrst elemen t of the co v erage p at tern b e indexed at 0. Num b er the subsets 0 thr ough r ( λ + 1). Deﬁne fun cti on C based on the co v erage patte rn , suc h that: C ( i ) =  σ if term i of the pattern is of v alue σ 0 otherwise Deﬁne fun cti on F on in tegers i , where 0 ≤ i ≤ r ( λ + 1): F ( i ) = P r − 1 j =0 C ( i + j − ∆). Deﬁne fun cti on F R on the same domain: F R ( i ) = F ( r ( λ + 1) − i ). The ﬁn al co v erage f unction deﬁned by the table is giv en b y F ( i ) + F R ( i ). The v alues that C ( i ) can tak e on are d epend en t on the t yp es of r u ns used. F or t yp e A r uns, C ( i ) can b e 0, 1/2 , or 1. Runs in tro duced later will ha ve a larger range of v alues, b ut it is alw a ys the ca se that 0 ≤ C ( i ) ≤ 1. Note that the functions F and F R giv en in CRE A TE-T ABLE- λ are piecewise linear functions with ranges dep endent on the fundament al pattern of co v erage. Du e to its constr u ctio n, the com bination F ( i ) + F R ( i ) is also a piecewise linear fun ctio n and symmetrical. Th us , only the r ange 0 ≤ i ≤ ⌊ r ( λ + 1) / 2 ⌋ n eed b e listed in tables. Although CREA TE-T ABLE- λ giv es a pro cedure for creati ng a table for a giv en sp eedup, w e need tables expressed sym b olically to prov e co v erage for a r an ge of sp eedups. Instead of usin g sp eciﬁc n um b ers, w e can lea v e the basic patterns of subset co v erage for a giv en st yle of run (suc h as t yp e A run s) with its shif ted version in terms of q and r . By shifting this pattern r times and summing the r esults together, w e accoun t for the diﬀerent alignmen ts a time window migh t h av e with resp ect to th e v arious su bin terv als. Th is sum is the function F , whic h can b e expressed as a piecewise linear function. F unction F R , wh ic h describ es reversed run s, can b e similarly describ ed. T o co mbine the t wo functions sym b olically , we sort the end p oin ts of the subset ranges from b oth descriptions together. If, for the giv en range of sp eedups b eing considered, there are tw o end p oints whic h cann ot b e ord ered, w e sub divide the range of sp eeds so that, in ea c h new sp eed range, the t w o end p oin ts in question can b e ordered. Once the end points of eac h sub set range ha v e b een sorted, com bining the descrip tions from the n ormal and rev ersed functions of the runs is ac hiev ed b y s imply summing eac h range. W e giv e an example of this pro cess in S ect . 5.1. 5 Sp eedup P erformance for Windo ws in S et W 2 Recall that W 2 is the set of w in do ws that completely ﬁll exactly t w o p erio ds. W e will no w explore the sp eedup-p erformance trade-oﬀ for win do ws in W 2 for all sp eedu ps 1 ≤ s ≤ 5. F or set W 2 , our 9 analysis m us t consider sub s et s w 0 through w 3 r . Th roughout our analysis, we will w e assign a 1 for full co v erage and a 1 / 2 f or half co v erage of an y subset. When examining the sub s et s for a giv en range of sp eedup v alues, the v alues are sym metrica l around w 3 r/ 2 when r is eve n and symmetrical after w (3 r − 1) / 2 when r is o dd. Th us, the tables and pro ofs w e use will not list contributions for subset w i where i > ⌊ 3 r / 2 ⌋ , since th e con tribu tio n at w i in these higher ranges is the same as the corresp onding con tribution at w 3 r − i , by symmetry . 5.1 Sp eedup 1 ≤ s ≤ 2 for Windo ws in Set W 2 F or the range 1 ≤ s ≤ 2, we can r epresen t an y rational sp eedup s in the form s = ( r + k ) /r with in tegers r ≥ 1 and 0 ≤ k ≤ r . F or this analysis, we consider service runs A , A R , A r − k , A R r − k , A 2 r − k , and A R 2 r − k , noting that λ = 2. Similar to W 1 for 1 ≤ s ≤ 2, r un A co vers set S 0 w ell, ru n A R co v ers set S 3 w ell, and the r emaining four run s plug the holes left in the sp ott y co v erage of sets S 1 and S 2 . W e will pass ov er the simp ler case for A run s and u se A r − k runs to give an example of h o w w e construct symbolic co v erage tables. F or sp eedup s where 1 ≤ s ≤ 2 and λ = 2, t yp e A runs h av e a fu ndamen tal pattern of co v erage of r s ubsets co v ered ev ery p eriod follo wed by q − r = k subsets co v ered ev ery other p erio d. Ad justing f or the oﬀset of ∆ = r − k and m aking r shifted, this p att ern yields th e v alues for F ( i ) and F R ( i ) giv en in T able 2 . F ( i ) =          k + i 3 2 r − 1 2 k − 1 2 i 2 r − 1 2 k − i r − 1 2 i 0 ≤ i ≤ r − k r − k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 2 r F R ( i ) =          1 2 i − 1 2 r i − r − 1 2 k 1 2 i − 1 2 k 3 r + k − i r ≤ i ≤ r + k r + k ≤ i ≤ 2 r 2 r ≤ i ≤ 2 r + k 2 r + k ≤ i ≤ 3 r T able 2: Separate co ve rage f unctions for A r − k and A R r − k in W 2 when 1 ≤ s ≤ 2. Because F ( i ) + F R ( i ) is symmetric ab out i = 3 r / 2 if r is ev en and after i = (3 r − 1) / 2 if r is o dd, we are only interested in the range 0 ≤ i ≤ ⌊ 3 r / 2 ⌋ . In this range, the sub-ranges r ≤ i ≤ 2 r − k and 2 r − k ≤ i ≤ ⌊ 3 r / 2 ⌋ for F o v erlap w ith the sub -ranges r ≤ i ≤ r + k and r + k ≤ i ≤ ⌊ 3 r / 2 ⌋ for F R . When k ≤ r − k , then r − k ≤ r + k ≤ ⌊ 3 r / 2 ⌋ . I n that case, for r + k ≤ i ≤ ⌊ 3 r / 2 ⌋ , F ( i ) + F R ( i ) = (2 r − k / 2 − i ) + ( i − r − k / 2) = r − k , as in the last in terv al of the middle set of con tributions in T able 3. When k ≥ r − k , then r − k ≤ 2 r − k ≤ ⌊ 3 r / 2 ⌋ . In that case, for 2 r − k ≤ i ≤ ⌊ 3 r/ 2 ⌋ , F ( i ) + F R ( i ) = ( r − i/ 2) + ( i/ 2 − r / 2) = r / 2, as in the last in terv al of the middle set of con tributions in T able 4. Similar analysis for runs A and A R and r uns A 2 r − k and A R 2 r − k pro duce the rest of T ables 3 and 4. The combined co ve rages of runs A , A R , A r − k , A R r − k , A 2 r − k , A R 2 r − k , and all o f their r espectiv e shifted versions are all giv en in T able 3 when k ≤ r − k and in T able 4 w h en k ≥ r − k . Lemma 5.1 If the c ontributions fr om A and A R ar e weighte d by a factor of 2 and the c ontributions fr om A r − k , A R r − k , A 2 r − k , and A R 2 r − k ar e weighte d by a factor of 1, the yield for al l intervals i s at le ast 2 r + k . 10 Com bined con tributions for A and A R =          r − 1 2 i r + 1 2 k − i 1 2 r + 1 2 k − 1 2 i 0 0 ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤  3 r 2  Com bined con tributions for A r − k and A R r − k =      k + i 3 2 r − 1 2 k − 1 2 i r − k 0 ≤ i ≤ r − k r − k ≤ i ≤ r + k r + k ≤ i ≤  3 r 2  Com bined con tributions for A 2 r − k and A R 2 r − k =                1 2 i i − 1 2 k 2 i − r + 1 2 k 3 2 i − 1 2 r + 1 2 k r + 2 k 0 ≤ i ≤ k k ≤ i ≤ r − k r − k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤  3 r 2  T able 3: Con tributions of run s for windo ws in W 2 when 1 ≤ s ≤ 2 and k ≤ r − k . Com bined con tributions for A and A R =          r − 1 2 i r + 1 2 k − i 1 2 r + 1 2 k − 1 2 i k − 1 2 r 0 ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤  3 r 2  Com bined con tributions for A r − k and A R r − k =      k + i 3 2 r − 1 2 k − 1 2 i 1 2 r 0 ≤ i ≤ r − k r − k ≤ i ≤ 2 r − k 2 r − k ≤ i ≤  3 r 2  Com bined con tributions for A 2 r − k and A R 2 r − k =                1 2 i 3 2 i − r + k 2 i − r + 1 2 k 3 2 i − 1 2 r + 1 2 k 5 2 r − k 0 ≤ i ≤ r − k r − k ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤  3 r 2  T able 4: Con tributions of run s for windo ws in W 2 when 1 ≤ s ≤ 2 and k ≥ r − k . Pr o of: W e ﬁrst consider the case wh en k ≤ r − k , consulting T able 3. If 0 ≤ i ≤ k , then the yield for w i is 2 r + k + i/ 2, which is at least 2 r + k , since i ≥ 0. If k ≤ i ≤ r − k , then th e yield f or w i is 2 r + 3 k / 2, which is greater than 2 r + k . If r − k ≤ i ≤ r , then the yield for w i is 5 r / 2 + k − i/ 2, whic h is at least 2 r + k , since i ≤ r . If r ≤ i ≤ ⌊ 3 r / 2 ⌋ , th en the yield for w i is 2 r + k . W e now consider the case when k ≥ r − k , consulting T able 4. If 0 ≤ i ≤ r − k , th en the yield for w i is 2 r + k + i/ 2, whic h is at least 2 r + k , since i ≥ 0. If r − k ≤ i ≤ k , then th e yield f or w i is 5 r / 2 + k / 2 , whic h is at least 2 r + k , s ince r ≥ k . 11 If k ≤ i ≤ r , then the yield for w i is 5 r / 2 + k − i/ 2, whic h is at least 2 r + k , sin ce i ≤ r . If r ≤ i ≤ 2 r − k , then th e yield f or w i is 2 r + k . If 2 r − k ≤ i ≤ ⌊ 3 r / 2 ⌋ , then the yield f or w i is also 2 r + k . ✷ Theorem 5.1 F or 1 ≤ s ≤ 2 , SP E EDUPW12 ﬁnds an 8 γ / ( s + 1) -appr oximation to the r e p airman pr oblem on windows in se t W 2 in O (min { r , m } Γ( n )) time. Pr o of: By Lemma 5.1, our analysis giv es no yield less than 2 r + k . S ince w e u se t wo copies ea ch of A and A R and a single cop y eac h of A r − k , A R r − k , A 2 r − k , and A R 2 r − k , a v eraged o v er r diﬀeren t sets of p eriod s, w e apply the Av erage Co v erage Pr oposition o v er 8 r runs. Thus, the fraction of optimal proﬁt obtained is (2 r + k ) / (8 γ r ) = (( r + k ) + r ) / (8 γ r ) = ( s + 1) / (8 γ ). ✷ 5.2 Sp eedup 2 ≤ s ≤ 3 for Windo ws in Set W 2 F or the r ange 2 ≤ s ≤ 5 / 2, w e can represen t an y rational sp eedup s in the f orm s = (2 r + k ) /r with inte gers r ≥ 1 and 0 ≤ k ≤ r − k . F or this analysis, we consider service runs A , A R , A r − 2 k , and A R r − 2 k , noting that λ = 2. W e will use three copies eac h of A and A R and a single cop y eac h of A r − 2 k and A R r − 2 k . Because the generation of the tables and the case analysis needed to sho w the co v erage are inv olv ed and of a similar form as Lemma 5.1, we ha ve mo v ed these d eta ils to App endix A. Theorem 5.2 F or 2 ≤ s ≤ 5 / 2 , SP EEDUPW12 ﬁnds an 8 γ / (2 s − 1) -appr oximation to the r ep air- man pr oblem on window s in set W 2 in O (min { r , m } Γ( n )) time. Pr o of: By Lemma A.1, the yield is at least 3 r + 2 k . Since three copies eac h of A and A R and a single cop y eac h of A r − 2 k and A R r − 2 k are used, av eraged o ve r r diﬀeren t sets of p erio ds, the Averag e Co v erage Prop osition is applied o v er 8 r ru ns. Thus, the fraction of optimal proﬁt obta ined is at least (3 r + 2 k ) / ( 8 γ r ) = ((4 r + 2 k ) − r ) / (8 γ r ) = (2 s − 1) / (8 γ ) . ✷ Observ at ion 5.1 Given a sp e e dup s ′ > s , we c an always simulate with sp e e dup s ′ the runs use d in the analysis of sp e e dup s by intr o ducing delays at appr opriate p oints in e ach run. Thus, an appr oximation r atio of β at sp e e dup s is an upp er b ound on the appr oximation r atio at sp e e dup s ′ . By Observ ation 5.1, the 2 γ -appro ximation f or s = 5 / 2 implies at m ost a constan t 2 γ -app ro ximation to the repairman pr oblem on windows in set W 2 when 5 / 2 ≤ s ≤ 3. 5.3 Sp eedup 3 ≤ s ≤ 5 for Windo ws in Set W 2 F or set W 2 with 3 ≤ s ≤ 5 where s = q /r , we consider runs A , A R , and their shifts, noting that λ = 2. When s < 4, run s A and ~ A giv e full co v erage for service requests in subsets of S 0 and S 1 and partial co v erage of service requests in S 2 , wh ile runs A R and ~ A R giv e full co v erage for s er v ice requests in subsets of S 3 and S 2 and partial co v erage of S 1 . When s = 4, runs A and ~ A go further b y also giving full cov erag e for service requests in s ubsets of S 2 , wh ile runs A R and ~ A R also giv e full co v erage for service requests in subsets of S 1 . When s > 4, r u ns A and ~ A giv e full co v erage for service requests in subsets of S 0 , S 1 , and S 2 and partial cov erag e of service r equests in S 3 , wh ile runs A R and ~ A R giv e full co v erage for service requests in sub s et s of S 3 , S 2 and S 1 and partial co v erage of S 0 . Since the con tributions of the A and A R runs and their shifted versions tend to balance eac h other, w e can analyze this balance b et w een the t wo ov er all p ossible sets of p er io ds to ﬁn d a lo w er b ound on the tot al pr oﬁ t after trimming. 12 Theorem 5.3 F or 3 ≤ s ≤ 5 , SPEEDUPW12 ﬁnd s a 4 γ / ( s − 1) -appr ox imation for windows W 2 in O (min { r , m } Γ( n )) time. Pr o of: F or ru ns A and ~ A , w i earns a 1 (denoting full co v erage) for eac h of the r sets of perio ds where 0 ≤ i ≤ q − 2 r , giving a total of r for eac h suc h i . F or eac h i > q − 2 r , the total decreases b y 1 / 2 fr om the to tal for i − 1. F o r ru ns A R and ~ A R , w i gets 1 for ea c h of the r sets of p eriod s where 3 r − ( q − 2 r ) ≤ i ≤ 3 r , giving a total of r for eac h suc h i . F or ea c h i < 3 r − ( q − 2 r ), the total decreases b y 1 / 2 from the tota l for i + 1. The com bined con tribu tio ns of runs A , A R , and their s hifted versions is q / 2 − r / 2 f or w 0 . Con tributions from ru n A are constan t, and con tributions from ru n A R only in crease or sta y constan t for w i where 0 < i ≤ r . Con tributions for w i for all runs sum to 2 r f or r < i < ⌊ 3 r/ 2 ⌋ . Th us , the yield for all w i is at least q / 2 − r / 2. Since tw o ru ns a ve raged ov er r diﬀeren t sets of p erio ds are used, the fraction of optimal pr oﬁ t obtained is at least ( q / 2 − r/ 2) · 1 / (2 γ r ) = q / (4 γ r ) − r / (4 γ r ) = ( s − 1) / (4 γ ). ✷ 6 New T yp es of Runs to Handle Windo ws in Set W 3 In a dd itio n to t yp e A run s, whic h rep eat every 2 p erio ds, our analysis of windo ws in W 3 deﬁnes t yp e B and typ e C runs whic h rep eat ev ery 3 or every 4 p erio ds, r esp ectiv ely . W e deﬁn e B and C runs only in the r ange 2 < s ≤ 3. Let ν = ⌈ s ⌉ − s . Start run B at t = 0 at the location that R ∗ has at time t = − 0 . 5. F rom there, run B follo ws a rep eating pattern of r aci ng forward along R ∗ for 2 p erio ds, racing bac kward along R ∗ for 1 − ν / (2 s ) p erio ds, and racing forward along R ∗ for ν / (2 s ) p erio ds. As for run C , also start it at t = 0 at the lo ca tion that R ∗ has at time t = − 0 . 5 . F rom there, run C follo ws a rep eating pattern of racing forw ard along R ∗ for 2 + 2 /s p erio ds and r ac ing bac kw ard along R ∗ for 2 − 2 /s p erio ds. Similar to A R , w e also deﬁne B R and C R , the “rev erses” of runs B and C , resp ectiv ely . Both runs B R and C R start at t = 0 at the lo catio n that R ∗ has at time t = λ/ 2. F rom its starting p oint, run B R follo ws a rep ea ting pattern of raci ng forw ard along R ∗ for ν / (2 s ), racing b ac kw ard along R ∗ for 1 − ν / (2 s ) p erio ds, and racing forw ard along R ∗ for 2 p erio ds. F rom its starting p oint , r un C R follo ws a rep eating pattern of racing bac kw ard along R ∗ for 2 − 2 /s p erio ds and forw ard along R ∗ for 2 + 2 /s p erio ds. As with A , let B ∆ and C ∆ , r esp ectiv ely , b e r u ns B and C mo v ed forw ard ∆ hops, and let run s B R ∆ and C R ∆ , resp ectiv ely , b e ru ns B R and C R mo v ed bac kw ard ∆ hops. Runs B ∆ and C ∆ , resp ectiv ely , follo w the same patterns of mo vemen t as ru n s B and C b ut start at t = 0 at the lo cation that R ∗ has at t = − 0 . 5 + ∆ / (2 r ). Their rev erses B R ∆ and C R ∆ , resp ectiv ely , follo w the same patterns of mo v emen t as B R and C R but sta rt at t = 0 at the location that R ∗ has at t = λ/ 2 − ∆ / (2 r ). As there is f or typ e A runs , there are unique patterns of co v erage corresp onding to t yp e B and C runs and their rev erses. Recall that s = q /r . Note that, for analysis of B and C runs, w e c ho ose the smallest v alues of q and r suc h that q + r is ev en. Since the num b er of subsets is determined b y r , it is necessary for B and C runs to ha ve an ev en q + r in order to k eep the co v erage deﬁ ned in terms of complete rather than partial sub sets. A typ e B run mo ves forw ard du ring its ﬁrst t w o p erio ds of time th e same distance that an optimal r un mov es du ring 2 q su bin terv al s. During its third p erio d of time, it mo v es bac kw ard the same distance that an optimal r u n mo v es durin g q (1 − ν / (2 s )) s u bin terv als and then forw ard the same distance that an optimal run m o v es durin g q ν / (2 s ) subinterv als. Then, the pattern rep eats. W e recall the list of subsets: S 0 1 , S 0 2 . . . S 0 r , S 1 1 . . . S 1 r , S 2 1 . . . S 2 r , S 3 1 . . . S 3 r , S 4 1 . . . S 4 r . When 2 < s ≤ 3, ru n B , during the ﬁrst p erio d in its pattern, co v ers q successiv e subsets as it mov es 13 forw ard. In the second p erio d, it cov ers another q subsets mo ving forw ard. Finally , in its third p erio d, it co v ers (3 q − 3 r ) / 2 sub sets bac kwa rd but no n ew sub s et s forward, since the su bsets co ve red forw ard were already co ve red bac kwa rd . W e see that only th e ﬁrst p erio d in the pattern co v ers the ﬁ rst ( q − r ) / 2 subsets. T h en the ﬁrst and third p eriod in the pattern co v er the n ext ( q − r ) / 2 subsets. All thr ee p eriod s co ver the next r subsets. Th e second and third p erio ds co v er th e next q − 2 r sub sets, and only the second p eriod co vers the ﬁnal r subsets. T his pattern of co ve rage is represent ed as ( q − r ) / 2 rep etitio ns of 1/3 , ( q − r ) / 2 rep etitions of 2/3, r rep et itions of 1, q − 2 r rep etitions of 2/3, and r rep etitions of 1/3. Figure 2 giv es t w o examples of t yp e B runs for sp eedups in the ran ge 17 / 7 ≤ s ≤ 3, the only range for which our analysis will emplo y t yp e B runs . Ob serv e that the run for s = 5 / 2 uses the form 10 / 4 in order to conform with the r estricti on for our analysis that q + r must b e ev en. Unlik e A runs which r epeat ev ery tw o p erio ds, b oth th e ru ns in this ﬁgure arriv e at the same corresp onding p osition at the b eginning of ev ery third p erio d, n amely at times 0 , 1 . 5 , 3 , and so on. Portio ns of runs servicing requests in S 0 , S 1 , S 2 , S 3 , and S 4 or v arious subsets are iden tiﬁed: su bsets S 0 4 , S 1 3 , S 1 4 , S 2 1 , and S 2 2 mapping to qu arter p erio ds of R ∗ for s = 10 / 4 and subsets S 0 5 , S 1 4 , S 1 5 , S 2 1 , S 2 2 , S 3 3 , and S 4 1 mapping to a ﬁfth of a p erio d of R ∗ for s = 13 / 5. F o cusing on the example of s = 10 / 4 where q = 10 and r = 4, n ote that, dur in g a thr ee- p erio d section, subsets S 0 1 through S 0 3 are co v ered a single time, su bsets S 0 4 , S 1 1 , and S 1 2 are co v ered twice , subsets S 1 3 , S 1 4 , S 2 1 , and S 2 2 are co ve red all three times, subsets S 2 3 and S 2 4 are co v ered twice, and subsets S 3 1 through S 3 4 are co v ered a single time. This pattern of 3 r epetitions of 1/3, 3 r epetitions of 2/3, 4 rep etitio ns of 1, 2 rep etitions of 2/3, and 4 r epetitions of 1/3 exactly corresp onds to the r epeating pattern of subset cov erage describ ed in the previous paragraph . Run B for s = 5 2 = 10 4 Run B for s = 13 5 Optimal Run R ∗ − . 5 0 . 5 1 1 . 5 2 2 . 5 3 0 . 5 1 1 . 5 2 3 0 . 5 1 1 . 5 2 3 S 0 S 1 S 1 3 S 1 4 S 2 1 S 2 2 S 0 4 S 2 S 1 S 0 S 3 S 2 S 1 S 1 3 S 1 4 S 2 1 S 2 2 S 0 4 S 2 S 1 S 0 S 0 S 1 S 2 S 1 S 0 S 3 S 2 S 1 S 1 4 S 1 5 S 2 1 S 2 2 S 2 3 S 0 5 S 4 1 S 1 4 S 1 5 S 2 1 S 2 2 S 2 3 S 0 5 S 2 S 1 S 0 Figure 2: Examp les o f typ e B runs for t w o d iﬀeren t sp eedups in the range 17 / 7 ≤ s ≤ 3, namely at 5 / 2 and 13 / 5. A t yp e C run mo ves f orward d u ring its ﬁr st t w o p erio ds of time the same distance that an optimal run m ov es during 2 q sub in terv als. During its third p erio d of time, it mo v es forwa rd the same distance that an optimal run mo v es d uring 2 q /s s u bin terv als and then bac kward the same distance that an optimal r un mo v es dur ing (1 − 2 /s ) q sub in terv als. During its fourth p erio d of time, it mov es bac kw ard th e same d istance that an optimal ru n mo v es d uring q su bin terv al s. Then, the p att ern rep eats. When 2 ≤ s ≤ 5 / 2, run C , d uring the ﬁrst p erio d in its pattern, co vers q successive subsets as it mo v es forward. In its second p eriod, it co v ers another q sub sets mo ving f orw ard. In its third p erio d, it co v ers 2 r subsets forwa rd b ut no new subsets bac kwa rd. Finally , in its fourth p erio d, 14 it cov ers q subsets mo ving bac kward. W e see that only th e ﬁrst p erio d in the p att ern co v ers the ﬁrst r subsets. Then, the ﬁrst and fourth p erio d in the pattern co v er the next q − 2 r su bsets. Th e ﬁrst, second, and four th p erio ds co v er the next r subs ets. The second and four th p erio ds co v er th e next q − 2 r su bsets. Th e second, third, and fourth p eriods co v er the next 3 r − q subsets. On ly the seco nd and the third p eriod co v er the next q − 2 r subsets, and only the third p erio d co v ers th e ﬁnal r sub sets. T his pattern of co v erage is repr esen ted as r rep etitio ns of 1/4, q − 2 r rep etitio ns of 1/2, r rep etitions of 3/4, q − 2 r rep etitions of 1/2, 3 r − q rep etitions of 3/4 , q − 2 r r epetitions of 1/2, and r rep etitio ns of 1/4. Just as with t yp e A run s, we will use the patterns for B and C runs in conjun cti on with CREA TE-T ABLE- λ to construct tables sho wing b oun ds on a v erage co v erage. Figure 3 g ive s t w o exa mples of t yp e C runs for sp ee du ps in the range 2 ≤ s ≤ 17 / 7, th e only range for whic h our analysis will employ t yp e C ru ns. Obser ve that the run for s = 2 u s es the form 4 / 2 in order to conform w ith the restrictio n for our analysis that q + r m ust b e ev en. Unlike A and B runs with, resp ectiv ely , rep eating patterns of t w o and three p erio ds, b oth th e ru ns in this ﬁgur e arriv e at the same p osition at the b egi nn ing of ev ery fourth p erio d , namely at times 0, 2, an d so on. P ortions of runs servicing requests in S 0 , S 1 , S 2 , S 3 , and S 4 or v arious subsets are iden tiﬁed: no separate su bsets for s = 2 but subsets S 1 2 , S 1 3 , S 1 4 , S 1 5 , S 2 1 , S 3 1 , S 3 2 , S 4 1 , and S 4 2 mapping to a ﬁfth of a p erio d of R ∗ for s = 11 / 5. F o cusing on the example of s = 11 / 5 where q = 11 and r = 5, note that, during a f our-p eriod section, subsets S 0 1 through S 0 5 are cov ered a single time, su bset S 1 1 is co v ered t wice, s ubsets S 1 2 through S 1 5 and S 2 1 are co v ered three times, subset S 2 2 is co v ered t wice, subsets S 2 3 through S 2 5 and S 3 1 are co v ered three times, subs et S 3 2 is co v ered twice , and subsets S 3 3 through S 3 5 and S 4 1 and S 4 2 are co v ered a single time. This pattern of 5 rep etitions of 1/4 , 1 rep etition of 1/2, 5 rep etitio ns of 3/4, 1 r epetition of 1/2, 4 r ep etitions of 3/4, 1 rep etition of 1/2, and 5 r epetitions of 1/4 exactly corresp onds to the r epeating pattern of su bset co v erage describ ed in the previous paragraph. Run C for s = 2 = 4 2 Run C for s = 11 5 Optimal Run R ∗ − . 5 0 . 5 1 1 . 5 2 2 . 5 3 0 . 5 1 1 . 5 2 2 . 5 0 . 5 1 1 . 5 2 2 . 5 S 0 S 1 S 1 S 2 S 2 S 1 S 0 S 3 S 2 S 1 S 2 S 0 S 1 S 2 1 S 1 2 S 1 3 S 1 4 S 1 5 S 2 S 3 1 S 3 2 S 2 3 S 2 4 S 2 5 S 1 S 0 S 3 S 2 S 1 S 4 1 S 4 2 S 3 1 S 2 1 S 1 2 S 1 3 S 1 4 S 1 5 Figure 3: Examp les o f typ e C ru ns for t w o diﬀerent sp eedup s in the range 2 ≤ s ≤ 17 / 7, namely at 2 and 11 / 5. 7 Sp eedup P erformance for Windo ws in S et W 3 W e will now explore the sp eedup-p erformance trade-oﬀ for w indo ws in W 3 for all sp eedups 1 ≤ s ≤ 6. F or set W 3 , our analysis m ust consider su b sets w 0 through w 4 r . As b efore, w e will assign a 1 for full cov erage and a 1 / 2 for half co v erage of any sub set. Because of the co v erage patterns of B and C r uns, w e w ill also assign v alues o f 1 / 4, 1 / 3, 2 / 3, and 3 / 4 for corresp onding prop ortions of co v erage. F or the subsets for a giv en range of sp eedup v alues for W 3 , the v alues are symmetrical around w 2 r . Thus, our tables and pro ofs will n ot list con tributions for sub set w i where i > 2 r . 15 7.1 Sp eedup 1 ≤ s ≤ 2 for Windo ws in Set W 3 F or this analysis, w e consider service r uns A , A R , A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , and A R 3 r − k , noting that λ = 3. W e will use t w o copies eac h of A and A R and a single copy eac h of A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , and A R 3 r − k . W e ha v e mo v ed the generation of th e tables and the case analysis needed to sho w the co v erage to App endix B. Theorem 7.1 F or 1 ≤ s ≤ 2 , our algorith m ﬁnds a 10 γ / ( s + 1) -appr oximation to the r ep airman pr oblem on windows in se t W 3 in O (min { r , m } Γ( n )) time. Pr o of: By Lemma B.1, our analysis giv es no yield less than 2 r + k . Since t w o copies eac h of A and A R and a single cop y eac h of A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , and A R 3 r − k are used, a v eraged o ver r diﬀeren t sets of p erio ds, th e Av erage Cov erage Prop osition is applied ov er 10 r runs. Th us , the fr ac tion of optimal proﬁt obtained is at lea st (2 r + k ) / (10 γ r ) = (( r + k ) + r ) / (10 γ r ) = ( s + 1) / (10 γ ) . ✷ 7.2 Sp eedup 2 ≤ s ≤ 7 / 3 for Windo ws in Set W 3 F or th e r ange 2 ≤ s ≤ 7 / 3 , an y rational sp eedup s can be represent ed in the form s = (2 r + k ) /r with in tegers r ≥ 1 and 0 ≤ k ≤ r/ 3. F or this analysis, we consider service r u ns A , A R , C (3 r − k ) / 2 , and C R (3 r − k ) / 2 , noting that λ = 3. W e w ill use a sin gle cop y eac h of A and A R and t w o copies eac h of C (3 r − k ) / 2 and C R (3 r − k ) / 2 . W e h a v e mo ved the generation of the tables and the case analysis needed to sho w the co v erage to App endix C. Theorem 7.2 F or 2 ≤ s ≤ 7 / 3 , algorithm SPE E DUPW12 ﬁnds a 6 γ /s -appr oxima tion to the r ep airman pr oblem on windows in set W 3 in O (min { r , m } Γ( n )) time. Pr o of: By Lemma C.1, our analysis giv es no yield less than 2 r + k . Since 1 cop y of eac h of A and A R and 2 co pies eac h of C (3 r − k ) / 2 and C R (3 r − k ) / 2 w ere used, a v eraged o v er r d iﬀeren t sets of p erio ds, the Av erage Co v erage P rop osition is applied o v er 6 r runs. Thus, the fr action of optimal proﬁt obtained is at least (2 r + k ) / (6 γ r ) = s/ (6 γ ). ✷ 7.3 Sp eedup 7 / 3 ≤ s ≤ 1 7 / 7 for Windo ws in Set W 3 F or the r ange 7 / 3 ≤ s ≤ 17 / 7, an y rational sp eedup s can b e r epresen ted in th e form s = (2 r + k ) /r with in tegers r ≥ 1 and r / 3 ≤ k ≤ 3 r / 7. F or this analysis, w e consider service r uns A , A R , C 2 r − 2 k , and C R 2 r − 2 k , noting that λ = 3. W e will us e k copies ea ch of A and A R and r − k copies eac h of C 2 r − 2 k and C R 2 r − 2 k . W e ha v e mo v ed generation of th e tables and th e case analysis n eeded to show the cov erag e to App endix D. Theorem 7.3 F or 7 / 3 ≤ s ≤ 17 / 7 , algorithm SPEEDUPW12 ﬁnds an 8 γ / ( s 2 − 4 s +7) -appr oximation to the r ep airman p r oblem on windows in set W 3 in O (min { r , m } Γ( n )) time. Pr o of: By Lemma D.1, our analysis giv es no yield less than 3 r 2 / 4 + k 2 / 4. Sin ce k copies of eac h of A and A R and r − k copies eac h of C 2 r − 2 k and C R 2 r − 2 k w ere used, a v eraged o ve r r diﬀeren t sets of p eriod s, the Av erage Co v erage Prop ositio n is app lie d o v er 2 r 2 runs. Th us, the fraction of optimal proﬁt obtained is at least (3 r 2 / 4 + k 2 / 4) / (2 γ r 2 ) = ((2 r + k ) 2 − 4 r (2 r + k ) + 7 r 2 ) / (8 γ r 2 ) = ( s 2 − 4 s + 7) / (8 γ ). ✷ 16 7.4 Sp eedup 17 / 7 ≤ s ≤ 3 for Windows in Set W 3 F or the ran ge 17 / 7 ≤ s ≤ 3, any rational sp eedup s can b e represent ed in the f orm s = (2 r + k ) /r with in tegers r ≥ 1 and 3 r / 7 ≤ k ≤ r . F or this analysis, we consider service run s A , A R , B r − k +1 , and B R r − k +1 , noting that λ = 3. W e will us e 6 r − 4 k copies eac h of A and A R and 3 r − 3 k copies eac h of B r − k +1 and B R r − k +1 . W e ha v e mov ed the generatio n of the tables and the case analysis needed to sho w the co v erage to App endix E. Theorem 7.4 F or 17 / 7 ≤ s ≤ 3 , algorithm SPEED UPW12 ﬁnds a γ (23 − 7 s ) / ( 1 + 3 s − s 2 ) - appr oximation to the r ep airman pr oblem on windows in set W 3 in O (min { r , m } Γ( n )) time. Pr o of: By L emma E.1, our analysis giv es no yield less than 6 r 2 − 2 r k − 2 k 2 . Sin ce 6 r − 4 k copies of eac h of A and A R and 3 r − 3 k copies eac h of B r − k +1 and B R r − k +1 w ere used, av erag ed o v er r diﬀeren t sets of p eriod s, the Av erage Co v erage Prop osition is applied o v er 18 r 2 − 14 r k runs. Th us , the fraction of optimal pr oﬁt obtained is at least (6 r 2 − 2 r k − 2 k 2 ) / ( γ (18 r 2 − 14 rk )) = ( r 2 + 3 r (2 r + k ) − (2 r + k ) 2 ) / ( γ r (23 r − 7(2 r + k ))) = (1 + 3 s − s 2 ) / ( γ (23 − 7 s )). ✷ By Observ ation 5.1, the 2 γ -appro ximation f or s = 3 implies at m ost a constan t 2 γ -appro ximation to the repairman pr oblem on windows in set W 3 when 3 ≤ s ≤ 4. 7.5 Sp eedup 4 ≤ s ≤ 6 for Windo ws in Set W 3 F or set W 3 with 4 ≤ s ≤ 6 where s = q /r , we consider runs A and A R , noting that λ = 3. When s < 5, run A and its shift giv e fu ll co v erage in subsets of S 0 , S 1 , and S 2 , and partial co v erage in subsets of S 3 and S 4 , wh ile ru n A R and its shift giv e f ull co verage in subsets of S 4 , S 3 , and S 2 and partial co v erage in sub sets of S 1 and S 0 . When s ≥ 5, run s A and ~ A giv e full co v erage in subsets of S 0 , S 1 , S 2 , and S 3 and p artial co v erage in subsets of S 4 , while ru ns A R and ~ A R giv e full co v erage in sub sets of S 4 , S 3 , S 2 , and S 1 and partial co v erage in subsets of S 1 and S 0 . S ince the con tributions of the A and A R runs and their shifted v ersions tend to balance eac h other, we can analyze this b ala nce b etw een the t wo o v er all p ossible sets of p erio ds to ﬁnd a lo we r b ound on the total pr oﬁt after trimming. Theorem 7.5 F or 4 ≤ s ≤ 6 , our algo rithm ﬁnds a 4 γ / ( s − 2) -app r oximation for windows W 3 in O (min { r , m } Γ( n )) time. Pr o of: A 1 is assigned f or an y su bset whic h is co v ered eve ry p erio d, an d a 1 / 2 is assigned f or an y sub set cov ered ev ery other p erio d. F or ru ns A and ~ A , w i earns a 1 for all r sets of p erio ds where 0 ≤ i ≤ q − 2 r , giving a total of r for eac h suc h i . F or eac h i > q − 2 r , the total decreases b y 1 / 2 from the total for i − 1. F or r uns A R and ~ A R , w i gets 1 for all r sets of p erio ds where 4 r − ( q − 2 r ) ≤ i ≤ 4 r , giving a total of r for eac h such i . F or eac h i < 4 r − ( q − 2 r ), the total decreases by 1 / 2 fr om the total for i + 1. No w, we tak e the total ov er all sets of p erio ds for b oth r u ns. F or runs A and ~ A , we get a total of r for w 0 . F or runs A R and ~ A R , we get a tota l of r − (1 / 2)(4 r − ( q − 2 r )) = q / 2 − 2 r . Summ in g these toge ther, w e get a yield of q / 2 − r for w 0 . By symmetry , the total for w 4 r is also q/ 2 − r . Con tributions from A and ~ A are constan t and con tributions from A R and ~ A R only increase or sta y constan t for 0 < i ≤ r . Contributions for w i for all runs sum to 2 r for r < i < 3 r . Th us, the yield for all other w i in all cases is at least q / 2 − r . S ince r sets of p erio ds for the t w o p airs o f runs cost a tota l of 2 r sets of p eriods to a v erage ov er, the fraction of proﬁt after trimming is at least ( q / 2 − r ) · 1 / (2 r ) = q / (4 r ) − 2 r / (4 r ) = ( s − 2) / 4. Multiplying the recipro cal b y γ giv es a 4 γ / ( s − 2)-appro ximation. ✷ 17 8 P erformance of S PEEDUPW12 No w that w e hav e c haracterized the p erformance of our sp eedup algorithms on w in do ws in sets W 1 , W 2 , and W 3 , w e b ound the p erform ance of S PEEDUPW12 b y com binin g our results as follo ws. Let R ∗ b e an optimal service run for a repairman instance with time windo w lengths from 1 u p to 2. Consid er a new set of p eriod s with duration . 25. Partit ion windo ws in to the sets H 3 , H 4 , H 5 , H 6 , and H 7 , such that for i = 3 , 4 , 5 , 6 , 7, a window is put in H i if it co mp letely con tains exactly i of these new p erio ds of length .25. Let the total fraction of pr oﬁt in an optimal solution coming from win d o ws in set H ℓ b e h ℓ . Thus, P 7 ℓ =3 h ℓ = 1. W e use these su bin terv a ls when analyzing the p erformance of the algorithm ru n on p erio ds of length . 5, . 75, and 1. Consider s et H ℓ of windows, ℓ = 3 , 4 , 5 , 6 , 7 and p erio d length j / 4, for j = 1 , 2 , 3 , 4. The n umber of full subin terv als of a windo w in H ℓ that are co v ered when the p erio d length is j / 4 is either ⌊ ( ℓ − j + 1) /j ⌋ or ⌈ ( ℓ − j + 1) /j ⌉ dep ending on whic h set of p erio ds is used. F or ℓ = 1 , 2 , 3, let f ℓ ( s ) /γ b e the fraction of optimal proﬁt earned for SPEEDUPW12 applied to requ ests with windo ws in W ℓ . Recall that co v erage of windows in W ℓ is deﬁn ed using p eriod size .5. T o app ly this co v erage to the three p erio d sizes used in the algorithm, we establish: Lemma 8.1 The ﬁrst pha se of SPEEDUPW12 yields a run ˆ R with γ · pr oﬁt ( ˆ R ) / pr oﬁt ( R ∗ ) = ρ 1 ≥ f 1 ( s ) h 3 +  1 2 f 1 ( s ) + 1 2 f 2 ( s )  h 4 + f 2 ( s ) h 5 +  1 2 f 2 ( s ) + 1 2 f 3 ( s )  h 6 + f 3 ( s ) h 7 . Pr o of: Windo ws from H 5 con tribute f 1 ( s ) h 3 in b oth sets of p erio ds. Windo ws from H 6 con tribute f 1 ( s ) h 4 in one set of p erio ds and f 2 ( s ) h 4 in the other. Windo ws from H 7 con tribute f 2 ( s ) h 5 in b oth sets of p eriod s. Windows from H 8 windo ws contribute f 2 ( s ) h 6 in one set of p erio ds and f 3 ( s ) h 6 in the other. Finally , wind o ws from set H 9 windo ws con tribute f 3 ( s ) h 7 in b oth sets of p erio ds. ✷ Lemma 8.2 The se c ond phase of SPEEDUPW 12 yields a run ˆ R with γ · pr oﬁt ( ˆ R ) / pr oﬁt ( R ∗ ) = ρ 2 ≥ 1 3 f 1 ( s ) h 3 + 2 3 f 1 ( s ) h 4 + f 1 ( s ) h 5 +  2 3 f 1 ( s ) + 1 3 f 2 ( s )  h 6 +  1 3 f 1 ( s ) + 2 3 f 2 ( s )  h 7 . Pr o of: Windows from H 5 con tribute 1 3 f 1 ( s ) h 3 in one set of p eriod s and nothing in the other t w o. Windo ws from H 6 con tribute 1 3 f 1 ( s ) h 4 in t wo sets of p erio ds and nothing in the other one. Windo w from H 7 con tribute 1 3 f 1 ( s ) h 5 in all three sets of p erio ds. Windo ws from H 8 con tribute 1 3 f 1 ( s ) h 6 in t w o sets of p eriods and 1 3 f 2 ( s ) h 6 in the other one. Windo ws from H 9 con tribute 1 3 f 1 ( s ) h 7 in one set of p eriods and 1 3 f 2 ( s ) h 7 in the other t wo . ✷ Lemma 8.3 The thir d phase of SP EEDUPW12 yields a run ˆ R with γ · pr oﬁt ( ˆ R ) / pr oﬁt ( R ∗ ) = ρ 3 ≥ 1 4 f 1 ( s ) h 4 + 1 2 f 1 ( s ) h 5 + 3 4 f 1 ( s ) h 6 + f 1 ( s ) h 7 . Pr o of: Windows from H 5 con tribute nothing in all four sets of p eriod s. Wind o ws from H 6 con- tribute 1 4 f 1 ( s ) h 4 in one set of p erio ds and nothing in the other three. Windo ws fr om H 7 con tribute 1 4 f 1 ( s ) h 5 in t w o sets of p erio ds and nothing in th e other tw o. Windo ws from H 8 con tribute 1 4 f 1 ( s ) h 6 in three sets of p erio ds and n ot hin g in the other one. Windo ws from H 9 con tribute 1 4 f 1 ( s ) h 7 in all four sets of p erio ds. ✷ F rom Lemmas 8.1, 8.2, and 8.3, w e isolat e the coeﬃcien ts b ℓ of the v a riables h ℓ , for ℓ = 3, 4, 5, 6, 7. W eight ing them b y x , y , and z to corr esp ond to those lemmas, resp ectiv ely , leads to the 18 follo wing deﬁnitions of ﬁ v e functions of x , y , z , and s . b 3 = f 1 ( s ) x + 1 3 f 1 ( s ) y b 4 =  1 2 f 1 ( s ) + 1 2 f 2 ( s )  x + 2 3 f 1 ( s ) y + 1 4 f 1 ( s ) z b 5 = f 2 ( s ) x + f 1 ( s ) y + 1 2 f 1 ( s ) z b 6 =  1 2 f 2 ( s ) + 1 2 f 3 ( s )  x +  2 3 f 1 ( s ) + 1 3 f 2 ( s )  y + 3 4 f 1 ( s ) z b 7 = f 3 ( s ) x +  1 3 f 1 ( s ) + 2 3 f 2 ( s )  y + f 1 ( s ) z Theorem 8.1 In O (min { r , m } Γ( n )) time, SPEEDUPW12 ﬁnds a run ˆ R such that γ · pr oﬁt ( ˆ R ) / ( pr oﬁt ( R ∗ )) ≥ max { ρ | ρ ≤ b ℓ for ℓ = 3 , 4 , 5 , 6 , 7 , x + y + z ≤ 1 , x ≥ 0 , y ≥ 0 , z ≥ 0 } . Pr o of: By Lemmas 8.1, 8.2, and 8.3, pr oﬁt ( ˆ R ) / ( p r oﬁt ( R ∗ ) γ ) ≥ max { ρ 1 , ρ 2 , ρ 3 } . Then , for any con v ex com bination of ρ 1 , ρ 2 , ρ 3 , (i.e., x, y , z ≥ 0 and x + y + z = 1), w e h a v e pr oﬁt ( ˆ R ) pr oﬁt ( R ∗ ) γ ≥ max x + y + z =1 x,y ,z ≥ 0 ( min ℓ ∈{ 3 , 4 , 5 , 6 , 7 } and h ℓ =1 { ρ 1 x + ρ 2 y + ρ 3 z } ) Th us , the expression for b ℓ is giv en by setting h ℓ = 1 and h i = 0, where i 6 = ℓ , and su mming ρ 1 , ρ 2 , and ρ 3 , w eigh ted by x , y , and z , resp ectiv ely . In this w a y , w e can acc ount for a problem instance b eing dominated by any set H ℓ for ℓ = 3 , 4 , 5 , 6 , 7. No p roblem instance will b e worse than a conv ex com bin at ion of all the b ounds. Algorithm SPEEDUPW12 runs SPEEDUP a tot al of 12 times in the ﬁrst phase, 6 times in the second p hase, and 4 times in the third phase, for a total of 22 times. W e hav e shown that the runn in g time of SPEEDUP is O (min { r , m } Γ( n )). ✷ W e giv e a description of f 1 ( s ), f 2 ( s ), and f 3 ( s ) in T able 5. F un cti on f 1 ( s ) come s from our w ork in [12]. F u n ctio n f 2 ( s ) comes from Sect. 5. F un ction f 3 ( s ) comes from Sect. 7. W e pro duced the results in T ab le 1 by solving the linear programs of Theorem 8.1 for p artic ular v alues of s w ithin eac h r an ge, inferring the patte rn for eac h range, and then pro ving the inferred pattern. Note th at all b ut one of the recipro cals of the r esulting ratios in terms of s are nonlinear functions! Theorem 8.2 F or sp e e dup s in the r ange 1 ≤ s ≤ 6 and window lengths b etwe en 1 and 2, algo- rithm SPEEDUPW12 pr o duc es a servic e run ˆ R for the r ep airman pr oblem with appr o ximation r atio pr oﬁt ( R ∗ ) / pr oﬁt ( ˆ R ) upp er-b ounde d as in T able 1 . Pr o of: F or eac h p ossible sp eedu p range, w e sh ow that γ times the con v ex com binations of the functions giv en in T able 5 are never less than the recipro cals of th e appr o ximati on ratios listed in T ab le 1. When 1 ≤ s ≤ 2, choose x = 50 73 , y = 6 73 , and z = 17 73 . Th en, b 3 = b 4 = . . . = b 7 = 26 s + 26 219 . When 2 ≤ s ≤ 7 / 3, choose x = 6 s 2 + 3 s 7 s 2 + 6 s + 3 , y = − 3 s 2 + 9 s 7 s 2 + 6 s + 3 , and z = 4 s s − 6 s + 3 7 s 2 + 6 s + 3 . 19 f 1 ( s ) ≥ ( ( s + 1) / 6 s/ 4 1 ≤ s ≤ 2 2 ≤ s ≤ 4 f 2 ( s ) ≥          ( s + 1) / 8 (2 s − 1) / 8 1 / 2 ( s − 1) / 4 1 ≤ s ≤ 2 2 ≤ s ≤ 5 2 5 2 ≤ s ≤ 3 3 ≤ s ≤ 5 f 3 ( s ) ≥                    ( s + 1) / 10 s/ 6 ( s 2 − 4 s + 7) / 8 (1 + 3 s − s 2 ) / (23 − 7 s ) 1 / 2 ( s − 2) / 4 1 ≤ s ≤ 2 2 ≤ s ≤ 7 3 7 3 ≤ s ≤ 17 7 17 7 ≤ s ≤ 3 3 ≤ s ≤ 4 4 ≤ s ≤ 6 T ab le 5: Lo w er b ounds on fractions of optimal proﬁt collecte d for the s ets W 1 , W 2 , and W 3 , ignoring the factor of γ . Then, b 3 = . . . = b 7 = 5 s 3 + 6 s 2 28 s 2 + 24 s + 12 . When 7 / 3 ≤ s ≤ 17 / 7, c ho ose x = 4 s 2 + 2 s − s 3 + 10 s 2 − 3 s + 2 , y = 3 s 3 − 18 s 2 + 27 s − s 3 + 10 s 2 − 3 s + 2 , and z = 4 s 3 − 24 s 2 + 32 s − 2 − s 3 + 10 s 2 − 3 s + 2 . Th en, b 3 = . . . = b 7 = s 4 − 2 s 3 + 11 s 2 − 4 s 3 + 40 s 2 − 12 s + 8 . When 17 / 7 ≤ s ≤ 5 / 2, c ho ose x = 14 s 3 − 39 s 2 − 23 s 17 s 3 − 43 s 2 − 35 s − 23 , y = − 9 s 3 + 54 s 2 − 81 s 17 s 3 − 43 s 2 − 35 s − 23 , and z = 12 s 3 − 58 s 2 + 69 s − 23 17 s 3 − 43 s 2 − 35 s − 23 . Th en, b 3 = . . . = b 7 = 11 s 4 − 21 s 3 − 50 s 2 68 s 3 − 172 s 2 − 140 s − 92 . When 5 / 2 ≤ s ≤ 3, choose x = 28 s 3 − 120 s 2 + 92 s 73 s 3 − 409 s 2 + 668 s − 368 , y = 33 s 3 − 189 s 2 + 264 s 73 s 3 − 409 s 2 + 668 s − 368 , and z = 12 s 3 − 100 s 2 + 312 s − 368 73 s 3 − 409 s 2 + 668 s − 368 . Th en, b 3 = . . . = b 7 = 39 s 4 − 183 s 3 + 180 s 2 292 s 3 − 1636 s 2 + 2672 s − 1472 . When 3 ≤ s ≤ 4, choose x = 2 s 2 + 2 3 s 2 + 2 s + 4 , y = − 3 s 2 + 12 s 3 s 2 + 2 s + 4 , and z = 4 s 2 − 12 s + 4 3 s 2 + 2 s + 4 . Then, b 3 = . . . = b 7 = s 3 + 6 s 2 12 s 2 + 8 s + 16 . When 4 ≤ s ≤ 5, choose x = 2 s − 2 − s + 16 , y = − 3 s + 18 − s + 16 , and z = 0. 20 Then, b 3 = b 7 = s + 4 − s + 16 , and b 4 = b 6 = s 2 − 6 s + 45 − 4 s + 64 > s + 4 − s + 16 , whenever s < 16. This follo ws since whenev er s < 16, s 2 − 6 s + 45 − 4 s + 64 > s + 4 − s + 16 holds if an d only if ( s − 5) 2 + 4 > 0, whic h is alw a ys true. Finally , b ound b 5 = s 2 − 8 s + 37 − 2 s + 32 > s + 4 − s + 16 whenev er s < 16. This follo ws since whenev er s < 16, s 2 − 8 s + 37 − 2 s + 32 > s + 4 − s + 16 holds if and on ly if ( s − 5) 2 + 20 > 0, whic h is alw a ys true. When 5 ≤ s ≤ 6, choose x = 8 − 3 s + 26 , y = − 3 s + 18 − 3 s + 26 , and z = 0. Then, b 3 = b 7 = s − 14 3 s − 26 , and b 4 = b 6 = 2 s − 20 3 s − 26 ≥ s − 14 3 s − 26 whenev er s ≤ 6. Finally , b 5 = 1 whic h is at least s − 14 3 s − 26 whenev er s ≤ 6. ✷ 9 Conclusion This pap er has demonstrated the sur prising v ersatilit y of the tec hniqu e of trimming. Even with time windo ws whose lengths are not all the same, it is p ossible to simp lify the structure of many time- constrained route-planning p r oblems and app ly an ord ering that allo ws dynamic programming to w ork w ell. F or unro oted problems, the cost of this add itio nal order is at most a constan t reduction in the pr oﬁt a run can earn. W e ha v e extended results fr om our previous pap er [12] so that w e can c haracterize the wa y in whic h this reduction in p roﬁt can b e oﬀset , in part or in wh ole , by sp eedup o v er a hypothetical optimal b enc hmark wh en the lengths of time windo ws are not all un iform. The k ey idea needed for this extension is to consider a div erse set of trials with a num b er of diﬀeren t p erio d lengths for trimming and th en c ho ose the b est resu lt among all those fou n d. T his app r oa ch mak es trimming adapt to v arious distributions of windo w lengths. W e ha ve give n tec hniqu es that ac hiev e an appro ximation ratio parameterized only b y sp eedup when th e ratio b etw een th e longest time wind ow and the shortest time win d o w is no greater than 2, b ut th ese tec hniques can b e extended to other ranges of time w indo w lengths. F or the general case, in whic h the ratio b et wee n the longest and th e shortest time wind o ws is D , th e appr oximati on ratio will worsen b y a factor of log 2 D , u s ing an app roac h similar to the one we used in [11] for general length time windows without sp eedup. It is w orth men tioning that we hav e ac hiev ed ap p ro ximation b ound s for a few sp eciﬁc ranges of s which are sligh tly b etter than the ones listed in T able 1. While trying to accommodate these ranges into a coheren t s c heme, our analysis became s o m uc h m ore co mplex that w e chose to giv e a more complete and readable presentat ion of results whic h are nearly as strong as the b est we found. The fact that b ett er v alues are p ossible sho ws that there is p oten tial in these tec hniques. References [1] E. M. Arkin, J. S. B. Mitc hell, and G. Narasimhan. Resource-constrained geomet ric net w ork optimization. In Pr o c. 14th Symp. on Computa tional Ge ometr y , pages 307–316, New Y ork, NY, USA, 19 98. A CM. 21 [2] N. Bansal, A. Blum, S. Cha wla, and A. Mey erson. Appro ximation algorithms for deadline-TSP and v ehicle routing with time-windo ws. In Pr o c. 36th ACM Symp. on The ory of Computing , pages 166–174, 2004 . [3] N. Bansal, H.- L. C h an, R. Khand ek ar, K. Pruhs, C. Stein, and B. Sc hieb er. Non-preemptiv e min-sum sc heduling with resour ce augmentat ion. In Pr o c. 48th IEEE Symp. on F oundations of Computer Scienc e , pages 614–624, W ashington, DC, USA, 2007 . IEEE C omputer So cie t y . [4] R. Bar-Y ehuda, G. Eve n, and S. Shahar. On appro ximating a geomet ric prize-colle cting tra v- eling salesman problem with time windows. J. A lgorithms , 55(1):76–92 , 2005 . [5] A. Blum, S. Cha wla, D. R. Karger, T. Lane, A. Mey erson, and M. Mi nkoﬀ. Approximat ion algorithms for orient eering and discounted-rew ard TSP. SIAM J . Comput. , 37(2):653–6 70, 2007. [6] C. C hekuri and N. Korula. Ap pro ximation algorithms for orien teering with time windows. 2007, http: //arxiv.org/a bs/0711.4825v1 . [7] C. Chekur i, N. Korula, and M. P´ al. Impro v ed algorithms for oriente ering and related p r oblems. In Pr o c. 19th ACM-SIAM Symp. on Discr ete Algorithms , p ag es 661– 670, Philadelphia, P A, USA, 2008. So cie ty for Ind ustrial and App lied Mathemati cs. [8] C. Chekuri and A. Kumar. Maxim um co ve rage p roblem with group bud get constraint s and ap- plications. In 7th Int. Workshop on Appr oxim ation A lgorithms for Combinatorial Optimization Pr oblems , v olume 3122 of LNCS , pages 72–83. Springer, 2004. [9] K. Chen and S. Har-P eled. The orien teering p roblem in the plane revisited. In Pr o c. 22nd Symp. on Computat ional Ge o metry , pages 247–254, New Y ork, NY, USA, 2006. A CM. [10] G. N. F rederickson an d B. Wittman. Appr o ximati on algorithms f or the tra v eling repairman and sp eeding deliv eryman pr oblems with un it-time windo ws. In APPROX-RANDOM , v olume 4627 of LNCS , pages 119–133. Springer, 2007. [11] G. N. F rederic kson and B. Wittman. Appro ximation algorithms for the tra ve ling re- pairman and sp eeding delive rym an problems. Jour n al v ersion, in s u bmission, a v aila ble: http://a rxiv.org/abs/0 905.4444 , 2009. [12] G. N. F rederickson and B. Wittman. Sp eedup in th e tra v eling repairman problem with unit time wind o ws. In submiss ion, a v ailable: ht tp://arxiv.org /abs/0907.5372 , 2009. [13] B. Kaly anasundaram and K. Pruhs. S p eed is as p o w erful as clai rvo y ance. J. A CM , 47(4):617– 643, 2000. [14] Y. Karuno, H. Nagamochi, and T. Ibaraki. Better appro ximation ratios f or the single-v ehicle sc heduling problems on line-shap ed net w orks. N etwo rks , 39(4):2 03–209, 2002. [15] V. Nagara jan and R. Ra vi. Po ly-logarithmic appro ximation algorithms for d irecte d v ehicle routing problems. In A PPR OX-RANDOM , v olume 462 7 of LNCS , pages 257– 270. Sp ringer, 2007. [16] C. A. Phillips, C. Stein, E. T orng, and J. W ein. Optimal time-critical sc heduling via resource augmen tation. Algorithmic a , 32(2):1 63–200, 2002. [17] J. N. Tsitsiklis. Sp ecial cases of trav eling salesman and repairman p roblems with time windo ws. Networks , 22:263–2 82, 19 92. 22 A Co v erag e of Wind o ws in Set W 2 when 2 ≤ s ≤ 3 Analysis of set W 2 when 2 ≤ s ≤ 3 is d on e b y considering s ervice runs A , A R , A r − 2 k , and A R r − 2 k , noting that λ = 2. T he com bined cov erag es of runs A , A R , A r − 2 k , A R r − 2 k , and all of their resp ectiv e shifted versions are giv en in T able 6 wh en k ≤ r − 2 k and in T able 7 when k ≥ r − 2 k . The com bined co v erage of the p air A and A R is exac tly the same for all v alues of k and are only listed in T able 6. These and all other tables in the App end ices are generated b y using CREA TE-T ABLE- λ for eac h diﬀeren t run t yp e, using the appropriate v alue of λ (2 for W 2 or 3 for W 3 ), and sp eciﬁc v alues for ∆ determined b y the n umb er of hops eac h run h as b een mo v ed. Com bined con tributions for A and A R =      r r + 1 2 k − 1 2 i 1 2 r + k 0 ≤ i ≤ k k ≤ i ≤ r − k r − k ≤ i ≤  3 r 2  Com bined con tributions for A r − 2 k and A R r − 2 k =          2 k + i 3 2 k + 3 2 i r + 1 2 i − 1 2 k 3 2 r − k 0 ≤ i ≤ k k ≤ i ≤ r − 2 k r − 2 k ≤ i ≤ r − k r − k ≤ i ≤  3 r 2  T ab le 6: Contributions of r uns for wind ows in W 2 when 2 ≤ s ≤ 5 / 2 and k ≤ r − 2 k . Com bined con tributions for A r − 2 k and A R r − 2 k =          2 k + i r r + 1 2 i − 1 2 k 3 2 r − k 0 ≤ i ≤ r − 2 k r − 2 k ≤ i ≤ k k ≤ i ≤ r − k r − k ≤ i ≤  3 r 2  T ab le 7: Contributions of r uns for wind ows in W 2 when 2 ≤ s ≤ 5 / 2 and k ≥ r − 2 k . Lemma A.1 If the c ontributions fr om A and A R ar e weighte d b y a factor of 3 and the c ontributions fr om A r − 2 k and A R r − 2 k ar e weighte d by a f actor of 1, the yield for al l intervals is at le ast 3 r + 2 k . Pr o of: W e ﬁrst consider the case wh en k ≤ r − 2 k , consulting T able 6. If 0 ≤ i ≤ k , then the yield for w i is 3 r + 2 k + i , which is at least 3 r + 2 k , since i ≥ 0. If k ≤ i ≤ r − 2 k , then the yield for w i is 3 r + 3 k , wh ich is greater than 3 r + 2 k . If r − 2 k ≤ i ≤ r − k , then the yield f or w i is 4 r + k − i , w hic h is at least 3 r + 2 k , sin ce i ≤ r − k . If r − k ≤ i ≤ ⌊ 3 r / 2 ⌋ , then the yield for w i is 3 r + 2 k . W e no w consider the ca se when k ≥ r − 2 k , consulting T ables 6 and 7. Th e algebra for the cases when 1 ≤ i ≤ r − 2 k , k ≤ i ≤ r − k , and r − k ≤ i ≤ ⌊ 3 r / 2 ⌋ giv es exactly the same results as the ﬁ rst, third, and fourth ranges from the pr evio us part of the pro of. If r − 2 k ≤ i ≤ k , then the yield for w i is 4 r ≥ 3 r + 2 k , sin ce r ≥ 2 k when 2 ≤ s ≤ 5 / 2. ✷ 23 B Co v erag e of Wind o ws in Set W 3 when 1 ≤ s ≤ 2 Analysis of set W 3 when 1 ≤ s ≤ 2 is done b y considerin g service ru ns A , A R , A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , and A R 3 r − k , n ot ing that λ = 3. The com bined cov erag es of ru n s A , A R , A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , A R 3 r − k , and all of their resp ectiv e s hifted versions are giv en in T able 8 when k ≤ r − k and in T able 9 wh en k ≥ r − k . The com bined co v erage s of the pair A and A R and the pair A r − k and A R r − k are exactly the same for all v alues of k and are only listed in T able 8. Com bined con tributions for A and A R =          r − 1 2 i r + 1 2 k − i 1 2 r + 1 2 k − 1 2 i 0 0 ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤ 2 r Com bined con tributions for A r − k and A R r − k =          k + i 3 2 r − 1 2 k − 1 2 i 2 r − 1 2 k − i r − 1 2 i 0 ≤ i ≤ r − k r − k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 2 r Com bined con tributions for A 2 r − k and A R 2 r − k =                0 i − r + k 3 2 i − 3 2 r + k 2 i − 2 r + 1 2 k 1 2 i + r − k 0 ≤ i ≤ r − k r − k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 2 r Com bined con tributions for A 3 r − k and A R 3 r − k =                1 2 i i − 1 2 k 1 2 i + 1 2 r − 1 2 k 2 r + k − i 2 k 0 ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 2 r T ab le 8: Contributions of r uns for wind ows in W 3 when 1 ≤ s ≤ 2 and k ≤ r − k . Lemma B.1 If the c ontributions f r om A and A R ar e weighte d by a factor of 2 and the c ont ributions fr om A r − k , A R r − k , A 2 r − k , A R 2 r − k , A 3 r − k , and A R 3 r − k ar e weighte d by a factor of 1, th e yield for al l intervals is at le ast 2 r + k . Pr o of: W e ﬁrs t consider th e case when k ≤ r − k , co nsu lting T able 8 . The algebra for the cases when 0 ≤ i ≤ r give s the same r esults the ﬁrst, second, and third cases in Lemma 5.1, at least 2 r + k in eac h case. If r ≤ i ≤ 2 r , then the yield for w i is 2 r + k . W e no w consider the case w hen k ≥ r − k , consulting T ables 8 and 9. Th e algebra for the cases when 0 ≤ i ≤ r gi ves the same results as the pr oof of Lemma 5.1 for k ≥ r − k , at lea st 2 r + k in eac h case. If r ≤ i ≤ 2 r , then the yield for w i is again 2 r + k . ✷ 24 Com bined con tributions for A 2 r − k and A R 2 r − k =                0 i − r + k 3 2 i − 3 2 r + k 3 2 r − 1 2 k 1 2 i + r − k 0 ≤ i ≤ r − k r − k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ r + k r + k ≤ i ≤ 2 r Com bined con tributions for A 3 r − k and A R 3 r − k =                1 2 i i − 1 2 k 1 2 i + 1 2 r − 1 2 k 3 2 i − 3 2 r + 1 2 k 2 k 0 ≤ i ≤ k k ≤ i ≤ r r ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ r + k r + k ≤ i ≤ 2 r T ab le 9: Contributions of r uns for wind ows in W 3 when 1 ≤ s ≤ 2 and k ≥ r − k . C Co v erage of Windo ws in Set W 3 when 2 ≤ s ≤ 7 / 3 Analysis of set W 3 when 2 ≤ s ≤ 7 / 3 is done b y considering service runs A , A R , C (3 r − k ) / 2 , and C R (3 r − k ) / 2 , noting that λ = 3. The combined co v erages for these ru ns are listed in T able 10 assum in g that r + k is ev en. When r + k is not ev en, we can ac hiev e an identical sp eed b y multiplying b oth b y 2. Com bined con tributions for A and A R =      r r + 1 2 k − 1 2 i k 0 ≤ i ≤ k k ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 2 r Com bined con tributions for C (3 r − k ) / 2 and C R (3 r − k ) / 2 =                1 2 r + 1 2 k + 1 2 i 3 4 r − 1 4 k 5 8 r − 1 8 k + 1 4 i 1 4 r + 1 4 k + 1 2 i r 0 ≤ i ≤ 1 2 ( r − 3 k ) 1 2 ( r − 3 k ) ≤ i ≤ 1 2 ( r − k ) 1 2 ( r − k ) ≤ i ≤ 1 2 (3 r − 3 k ) 1 2 (3 r − 3 k ) ≤ i ≤ 1 2 (3 r − k ) 1 2 (3 r − k ) ≤ i ≤ 2 r T ab le 10: Con tribu tio ns of ru ns for windows in W 3 when 2 ≤ s ≤ 7 / 3. Lemma C.1 If the c ontributions f r om A and A R ar e weighte d by a factor of 1 and the c ontributions fr om C (3 r − k ) / 2 and C R (3 r − k ) / 2 ar e weighte d by a factor of 2, the yield for al l intervals is at le ast 2 r + k . Pr o of: C onsulting T able 10, w e ﬁrst consid er th e case when 5 k ≤ r , whic h implies k ≤ ( r − 3 k ) / 2. If 0 ≤ i ≤ k , then the yield for w i is 2 r + 4 k + i , which is at least 2 r + k , since i ≥ 0. If k ≤ i ≤ ( r − 3 k ) / 2, then the yield for w i is 2 r + 3 k / 2 + i/ 2, whic h is greater than 2 r + k . If ( r − 3 k ) / 2 ≤ i ≤ ( r − k ) / 2, then the yield for w i is 5 r / 2 − i/ 2, w hic h is at least 2 r + k , since i ≤ ( r − k ) / 2 and r ≥ 3 k . If ( r − k ) / 2 ≤ i ≤ (3 r − 3 k ) / 2 , then the yield for w i is 9 r/ 4 + k / 4, whic h is at least 2 r + k , since r ≥ 3 k . 25 If (3 r − 3 k ) / 2 ≤ i ≤ (3 r − k ) / 2, then the yield for w i is 3 r / 2 + k + i/ 2, wh ic h is at least 2 r + k , since i ≥ (3 r − 3 k ) / 2 and r ≥ 3 k . If (3 r − k ) / 2 ≤ i ≤ 2 r − k , then the yield for w i is 3 r + k / 2 − i/ 2, whic h is at lea st 2 r + k , since i ≤ 2 r − k . If 2 r − k ≤ i ≤ 2 r , then the yield for w i is 2 r + k . Next, we consider the case wh en 5 k ≥ r , again consu lting T able 10. The ﬁr st case giv es the same r esult as ab o v e but for th e range 0 ≤ i ≤ ( r − 3 k ) / 2. If ( r − 3 k ) / 2 ≤ i ≤ k , then the yield for w i is 5 r / 2 − k / 2, whic h is at least 2 r + k , since r ≥ 3 k . The third case giv es the same result as ab o ve but for the range k ≤ i ≤ ( r − k ) / 2. The fourth, ﬁfth, and sixth cases abov e giv e iden tical results when 5 k ≥ r . ✷ D Co v erage of Windo ws in Set W 3 when 7 / 3 ≤ s ≤ 17 / 7 Analysis of set W 3 when 7 / 3 ≤ s ≤ 17 / 7 is done b y consid ering service r uns A , A R , C 2 r − 2 k , and C R 2 r − 2 k , noting that λ = 3. Th e combined co v erages for r uns A and A R are listed in T able 10 , and the com bined co ve rages for r uns C 2 r − 2 k and C R 2 r − 2 k are listed in T able 11, assuming that r + k is ev en. Com bined con tributions for C 2 r − 2 k and C R 2 r − 2 k =                    3 4 r − 1 4 k 1 2 r + 1 4 k + 1 4 i 1 4 r + 1 4 k + 1 2 i 3 4 r + 3 4 k 1 4 r + k + 1 4 i 1 2 r + 3 2 k 0 ≤ i ≤ r − 2 k r − 2 k ≤ i ≤ r r ≤ i ≤ r + k r + k ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ r + 2 k r + 2 k ≤ i ≤ 2 r T ab le 11: Contributions of runs for windows in W 3 when 7 / 3 ≤ s ≤ 17 / 7. Lemma D.1 If the c ontr ibutions fr om A and A R ar e weighte d by a factor of k and the c ontributions fr om C 2 r − 2 k and C R 2 r − 2 k ar e weighte d by a factor of r − k , the yield for al l intervals is at le ast 3 r 2 / 4 + k 2 / 4 . Pr o of: Consu lt T ables 10 and 11. If 0 ≤ i ≤ r − 2 k , then the yield f or w i is 3 r 2 / 4 + k 2 / 4. If r − 2 k ≤ i ≤ k , then the yield for w i is r 2 / 2 + 3 r k / 4 + ( r − k ) i/ 4, whic h is greater than 3 r 2 / 4 + k 2 / 4, sin ce i ≥ r − 2 k . If k ≤ i ≤ r , then the yield for w i is r 2 / 2 + 3 r k/ 4 + k 2 / 4 + ( r − 3 k ) i/ 4, wh ic h is at least 3 r 2 / 4 + k 2 / 4, sin ce i ≤ r and r ≤ 3 k . If r ≤ i ≤ r + k , then the yield for w i is r 2 / 4 + r k + k 2 / 4 + ( r − 2 k ) i/ 2, which is at least 3 r 2 / 4 + k 2 / 4, sin ce i ≥ r . If r + k ≤ i ≤ 2 r − k , then the yield for w i is 3 r 2 / 4 + rk − k 2 / 4 − ki/ 2, wh ic h is at least 3 r 2 / 4 + k 2 / 4, sin ce i ≤ 2 r − k . If 2 r − k ≤ i ≤ r + 2 k , then the yield for w i is r 2 / 4 + 3 r k/ 4 + ( r − k ) i/ 4, whic h is at least 3 r 2 / 4 + k 2 / 4, sin ce i ≥ 2 r − k . If r + 2 k ≤ i ≤ 2 r , then the yield for w i is r 2 / 2 + r k − k 2 / 2, whic h is at least 3 r 2 / 4 + k 2 / 4, since ( r 2 / 2 + r k − k 2 / 2) − (3 r 2 / 4 + k 2 / 4) = (( r − k ) / 2)((3 k − r ) / 2) ≥ 0 when k ≤ r ≤ 3 k . ✷ 26 E Co v erage of Windo ws in Set W 3 when 17 / 7 ≤ s ≤ 3 Analysis of set W 3 when 17 / 7 ≤ s ≤ 3 is done b y considering ser v ice r uns A , A R , B r − k +1 , and B R r − k +1 , noting that λ = 3. T he com bined co v erages for run s A and A R are listed in T able 10, and the com bin ed co v erages for runs B r − k +1 and B R r − k +1 are listed in T able 12 when 17 / 7 ≤ s ≤ 5 / 2 and in T able 13 when 5 / 2 ≤ s ≤ 3, assum ing in b oth cases that r + k is ev en. Com bined con tributions for B r − k +1 and B R r − k +1 =                    2 3 k + 2 3 i − 1 3 r + k + i − 1 2 r + 5 6 k + 4 3 i 1 3 k + i 2 3 r + 2 3 i 5 3 r + 1 3 k 0 ≤ i ≤ r − k r − k ≤ i ≤ 1 2 ( r + k ) 1 2 ( r + k ) ≤ i ≤ 1 2 (3 r − 3 k ) 1 2 (3 r − 3 k ) ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 1 2 (3 r + k ) 1 2 (3 r + k ) ≤ i ≤ 2 r T ab le 12: Contributions of runs for windows in W 3 when 17 / 7 ≤ s ≤ 5 / 2. Com bined con tributions for B r − k +1 and B R r − k +1 =                    2 3 k + 2 3 i − 1 3 r + k + i 1 6 r + 1 2 k + 2 3 i 1 3 k + i 2 3 r + 2 3 i 5 3 r + 1 3 k 0 ≤ i ≤ r − k r − k ≤ i ≤ 1 2 (3 r − 3 k ) 1 2 (3 r − 3 k ) ≤ i ≤ 1 2 ( r + k ) 1 2 ( r + k ) ≤ i ≤ 2 r − k 2 r − k ≤ i ≤ 1 2 (3 r + k ) 1 2 (3 r + k ) ≤ i ≤ 2 r T ab le 13: Con tribu tio ns of ru ns for windows in W 3 when 5 / 2 ≤ s ≤ 3. Lemma E.1 If the c ontributions fr om A and A R ar e weighte d by a factor of 6 r − 4 k and the c ontr ibutions fr om B r − k +1 and B R r − k +1 ar e weighte d by a factor of 3 r − 3 k , the yield for al l intervals is at le ast 6 r 2 − 2 r k − 2 k 2 . Pr o of: In the case that 17 / 7 ≤ s ≤ 5 / 2, consult T ables 10 and 12. If 0 ≤ i ≤ k , then the yield for w i is 6 r 2 − 2 r k − 2 k 2 + 2( r − k ) i , whic h is at least 6 r 2 − 2 r k − 2 k 2 , since r ≥ k . If k ≤ i ≤ r − k , then the yield for w i is 6 r 2 + r k − 4 k 2 − ir = 6 r 2 − 2 r k − 2 k 2 + ( r − k − i ) r + ( r − 2 k ) k + (3 k − r ) r , whic h is greater than 6 r 2 − 2 r k − 2 k 2 , since i ≤ r − k , r ≥ 2 k , and k > r / 3. If r − k ≤ i ≤ ( r + k ) / 2 , then the yield for w i is 5 r 2 + 3 r k − 5 k 2 − ik = 6 r 2 − 2 r k − 2 k 2 + (( r + k ) / 2 − i ) k + ( r − 2 k )7 k / 4 + (7 k − 3 r ) r / 3 + 5 r k / 12 , whic h is greater than 6 r 2 − 2 r k − 2 k 2 , since i ≤ ( r + k ) / 2, r ≥ 2 k , and k ≥ 3 r / 7. If ( r + k ) / 2 ≤ i ≤ (3 r − 3 k ) / 2, then the yield for w i is 9 r 2 / 2 + 3 r k − 9 k 2 / 2 + ( r − 2 k ) i = 6 r 2 − 2 r k − 2 k 2 + ( i − ( r + k ) / 2)( r − 2 k ) + ( r − 2 k )7 k/ 4 + (7 k − 3 r ) r / 3 + 5 r k / 12, wh ic h is greater than 6 r 2 − 2 r k − 2 k 2 , sin ce i ≥ ( r + k ) / 2 , r ≥ 2 k , and k ≥ 3 r / 7. If (3 r − 3 k ) / 2 ≤ i ≤ 2 r − k , then the yield for w i is 6 r 2 − 3 k 2 − ik , whic h is at least 6 r 2 − 2 r k − 2 k 2 , since i ≤ 2 r − k . 27 If 2 r − k ≤ i ≤ (3 r + k ) / 2, then the yield for w i is 2 r 2 + 4 r k − 4 k 2 + 2( r − k ) i , whic h is at least 6 r 2 − 2 r k − 2 k 2 , since i ≥ 2 r − k . If (3 r + k ) / 2 ≤ i ≤ 2 r , then the yield for w i is 5 r 2 + 2 r k − 5 k 2 = 6 r 2 − 2 r k − 2 k 2 + ( r − 2 k )3 k / 2 + (7 k − 3 r ) r / 3 + r k/ 6, whic h is at least 6 r 2 − 2 r k − 2 k 2 , sin ce r ≥ 2 k and k ≥ 3 r / 7. In the case that 5 / 2 ≤ s ≤ 3, consult T ables 10 and 13. F or this range (3 r − 3 k ) / 2 ≤ ( r + k ) / 2. If r − k ≤ i ≤ (3 r − 3 k ) / 2, then the yield for w i is 5 r 2 + 3 r k − 5 k 2 − ik = 6 r 2 − 2 r k − 2 k 2 + ((3 r − 3 k ) / 2 − i ) k + ( r − k )3 k / 2 + (2 k − r ) r , which is at least 6 r 2 − 2 r k − 2 k 2 , since i ≤ (3 r − 3 k ) / 2 and k ≤ r ≤ 2 k . If (3 r − 3 k ) / 2 ≤ i ≤ ( r + k ) / 2, then the yield for w i is 13 r 2 / 2 − 7 k 2 / 2 − ir , whic h is at least 6 r 2 − 2 r k − 2 k 2 , since i ≤ ( r + k ) / 2 and r ≥ k . If ( r + k ) / 2 ≤ i ≤ 2 r − k , then the yield for w i is 6 r 2 − 3 k 2 − ik , wh ic h is at least 6 r 2 − 2 r k − 2 k 2 , since i ≤ 2 r − k . All other ranges are id en tical to s ome yield wh en 17 / 7 ≤ s ≤ 5 / 2. ✷ 28

Speedup in the Traveling Repairman Problem with Constrained Time Windows

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment