Two Multivehicle Routing Problems with Unit-Time Windows

Tw o Multiv ehicle Routing Pro blems with Unit-Time Windo ws Greg N. F rederic kson ∗ Barry Wittman † Septem b er 28, 201 8 Abstract Two m ultivehicle routing problems are considered in the framework that a visit to a lo cation m ust take place during a sp ecific time window in order to b e counted a nd all time windows a r e the same length. In the first problem, the goal is to visit as man y loca tions as possible using a fixed num b er of vehicles. In the sec ond, the g oal is to visit a ll loca tions using the sma lle st nu mber of vehicles possible. F or the first problem, we present an approximation algorithm whose output path collects a rew ar d within a consta n t fac to r of optimal for any fixed num ber of vehicles. F or the s econd pro blem, o ur alg o rithm finds a 6 -approximation to the pro blem on a tr e e metric, whenever a single vehicle could visit all lo cations dur ing their time windows. Key wor ds: Appr oximation algor ithms, analysis of algorithms, g raph alg orithms 1 In tro duction In [6] w e introdu ced p olynomial time, constan t-facto r approxima tion alg orithms to the T rav el ing Repairman Problem w ith unit-time windows. In that pr oblem, a single agen t tra v eling at a fixed sp eed on a w eigh ted graph m ust visit as many locations as p ossible during their time windo ws. Although th is is a pr oblem general enough to mo del man y practical r outing and p ath-planning problems, it only considers a single agent . Man y commercial indu stries h av e large fleets of v ehicles whose r outes m ust b e co ordinated together for maximum efficiency . In this pap er we in tro du ce approximat ion algorithms for some multiv ehicle p roblems, often called v ehicle routing problems. O ur work in this area, naturally , fo cuses on the addition of time windo ws to v ehicle routing pr oblems without time constrain ts. The v ehicle routing problem was first in tro d u ced in [4], w hic h fr ames the problem w ith a cen tral dep ot, a set of a tr uc ks, and a set of lo cations requiring pro d uct from the dep ot. The goal of this prob lem is to minimize the total mileage trav eled by the fl eet of tru c ks in servicing all lo cations. O ur v ersion of th e prob lem assumes that no time is required at the lo cation to p erform the service b ecause service times can easily b e absorb ed in to the stru ctur e of the graph [7]. As with single v ehicle p roblems, the addition of time constraints in tro duces seve ral d ifferen t kinds of optimization b ecause it ma y no longer b e p ossible to service all lo cations with a give n fleet of v ehicles and hard time constraints. In [5], the authors find a constant app r o ximation for the problem in wh ic h there are n o hard time constraint s, b ut th e goal is to minimize the av erage time customers ha v e to wa it. In [9], only a release time is giv en for eac h lo cation, and the goal is to minimize the maximum lateness that an y lo cation is serviced after its release time. F or this problem, a PT AS is found, bu t only when the lo cations are on a p ath. Th e auth ors of [8] similarly “soften” ∗ Dept. of Computer S ciences, Purdue Universit y , W est Lafay ette, IN 47907. gnf@cs.purdu e.edu † Dept. of Computer S cience, Elizabethtow n College, Elizab ethto wn, P A 17022. wi ttmanb@etown.ed u 1 time wind o w r equiremen ts by changing lateness in to a cost function to b e minimized. Th e goal of the orien teering pr oblem is to visit as many lo cations as p ossible b efore a global deadline. I n [2 ], an algorithm is giv en that uses an α -app ro ximation to orien teering to fi nd an ( α + 1)-appro ximation for the m ultiple-path orien teering p roblem, for whic h k different vehicles are trying to maximize the total n umb er of lo cations visited at least once b efore a global deadline. With the in tro duction of multiple ve hicles, it is reasonable to try to minimize th e num b er of v ehicles used. In [1], th e goal is to minimize the total num b er of ve hicles needed to service all lo cations, starting and returning to a dep ot, with a hard constraint D on the total distance a single v ehicle can tra v el. T h is distance limitation can b e viewed as a global deadline. A similar pr oblem, with the requir ement of returnin g to the dep ot remo ved, is discussed in [10] w h ere a 4-appro ximation is give n for tree metrics and an O (log D )-appro ximation is giv en f or general metrics. In Section 2, we defin e the T rav eling Rep airman Problem from [6]. W e will use appro ximation algorithms for this problem as s u broutines. More imp ortan tly , we w ill explain the concept of trimming that pla ys an inte gral r ole in b oth the algorithms in [6] and in this p ap er. In Section 3, f or any fi xed k w e giv e a constan t-factor p olynomial-time approxima tion algorithm for the unr o oted k -v ehicle routing p roblem with unit-time win do ws. This problem is unr o ote d in the sense that v ehicles are p ermitted to start at an y time at an y lo cation and n eed not return to a cen tral dep ot by a deadline. In Section 4, we consider a second problem, namely the Minimum V ehicle O P T = 1 Pr oblem discussed in [10]. In th is pr oblem, it is assumed that a single vehicle is enough to service all lo cations b efore a global d eadline. Ou r app ro ximation algorithm find s paths that guaran tee that some small n umb er of v ehicles is suffi cient to service eve ry lo cation b efore th e deadline. A 14-appro ximation for this problem is giv en in [10]. W e c hange th is problem from a sin gle deadline to time wind o ws in the unro oted setting and giv e a 6-appr o ximation to this pr oblem on tree metrics. Most notable in this algorithm is the replacemen t of trimming by expansion, extending time win do ws to b e longer rather than s horter as with the usual compression. In these s ections, we p resen t app r o ximation algorithms for t w o fun damen tal m ultiv ehicle p rob- lems. Both the form of the problems and our approac hes to solving them offer in teresting con trasts. While th e multiv ehicle routing p roblem u ses a simple algo rithm that rep eatedly runs a single v ehicle repairman appr o ximation, the analysis needed to show the p erformance of this algorithm requires a ju dicious c hoice of b oun ding terms. On the other h and, the minimum v ehicle approxi mation algorithm we presen t dep ends on a clev er reversal of the trimming tec hn iqu es from [6], b u t the analysis is im m ediate. 2 Repairman and T rimming W e will u se approxi mation algorithms for the T rav eling Repairman Problem [6], a 1-v ehicle v ersion of the p r oblems consider ed in this p ap er. In this problem, a repairman is presente d w ith a set of servic e r e q uests . Eac h serv ice request is lo cated at a n o de in a we igh ted, un directed graph and is assigned a time window during whic h it is v alid. Th e goal of the r epairman is to plan a route called a servic e run that services as man y requests as p ossible du r ing their time windows while trav eling at a giv en fixed sp eed. Note that w e consider th e unro oted v ersion of th e problems, in which the agen t ma y start at any time fr om any lo cation and stop similarly . W e giv e appro ximation algorithms in [6] that guarantee a 3 γ -appr o ximation to the T ra ve ling Repairman P r oblem with unit-time windows and run in Γ( n ) time. F or a tree, γ = 1 and Γ( n ) is O ( n 4 ). F or a m etric graph , γ = 2 + ǫ and Γ( n ) is O ( n O (1 /ǫ 2 ) ), w ith the addition of improv emen ts to [6 ] give n in [3]. A more thorough explanation of these improv emen ts is a v ai lable in [7]. 2 T o ac hieve these results, we use a tec hnique called trim m ing that is effectiv e when we deal with unit-time windo ws. Starting with time 0, w e make d ivisions in time at v a lues whic h are in teger m ultiples of one half, i.e., 0, .5, 1, and so on. W e assume that no r equest windo w s tarts on su ch a division, b ecause w e can alwa ys redefine times to b e decreased by a n egligible amount. W e thus assume that the s tarting time for any w in do w is p ositive . Let a p erio d b e th e time interv al from one division up to b u t not including the n ext d ivision. Beca use ev ery service r equ est h as a time windo w exactly one u nit long, half of that time win d o w w ill b e wholly con tained within just one p erio d, with th e rest of the time win do w divided b et w een the p receding and follo wing p erio ds. W e then trim eac h serv ice r equest window to coincide pr ecisely with the p erio d wh olly con tained in it, ignoring those p ortions of the request window that fall outside of the c hosen p eriod . F or the Repairman Problem, the tr imming may we ll lo w er the pr ofi t of the b est service run, but b y no more than a factor of 3, as giv en in the Limited Loss Theorem of [6]. 3 Multiv ehicle Problem with Unit-Time Windo ws In this section w e define the Multivehicle R outing Pr oblem and giv e a constan t app ro ximation to the unit-time window case when there are tw o vehicl es. W e sho w ho w this approac h can b e extend ed to k ve hicles at the p rice of sacrificing a constant appro ximation. W e define the Multiv ehicle Routing Problem to tak e the same input as the T ra v eling Repairman Problem, b ut with a w h ole n umber sp ecifying the num b er of v ehicles. The goal is to assign a service run for eac h ve hicle such that the total profit is maximized. As b efore, a service request can only b e serviced once, and it m ust b e serviced d uring its sp ecified time windo w. W e assume th at the time windo ws are all of unit length. F or the case of t w o vehicle s, w e run the algo rithm called 2VEHICLE: First divide the graph in to p erio ds of .5 time units and trim requests into those p erio d s as explained in Section 2. Next, run our single vehicle rep airm an appro ximation from [6 ] on the trimmed requests and call the resulting service r un R 1 . Then, remov e all th e requests serviced b y the first pass of our single v ehicle appr o ximation, run the appro ximation a second time, and call the resulting service ru n R 2 . Runs R 1 and R 2 are the output of 2VEHICLE. Let p ( R ) giv e the pr ofit collected b y ru n R for servicing lo cations, and let c ( R ) giv e the c ost of trav eling along run R und er metric d . Let ser v ice r uns R ∗ 1 and R ∗ 2 service d isjoin t subsets of r equests and b e optimal in the sense th at the quan tit y p ( R ∗ 1 ) + p ( R ∗ 2 ) is maximized. W.l.o.g., let p ( R ∗ 1 ) ≥ p ( R ∗ 2 ). Service ru ns R 1 and R 2 ma y o v erlap arbitrarily with service run s R ∗ 1 and R ∗ 2 . Let p ∗ 1 ( R 1 ) b e the pr ofits earned by R 1 b y servicing r equests wh ic h were serviced b y run R ∗ 1 . Similarly , let p ∗ 2 ( R 1 ) b e the p rofits earned b y R 1 b y servicing requests whic h w ere serviced by ru n R ∗ 2 . Define p ∗ 1 ( R 2 ) and p ∗ 2 ( R 2 ) in a similar manner. Lemma 3.1 Algorith m 2VEHICLE gives a 36 γ 2 12 γ − 1 -appr oximation to the Multivehicle R outing Pr ob- lem with u ni t- time windows for 2 vehicles on any metric gr aph . Pr o of: The profit collected by run R ∗ 1 cannot b e greater than the pr ofit colle cted b y a single ve hicle optimal tour. T h us, by our earlier argum ents ab out the impact of trimming, p ( R 1 ) ≥ 1 3 γ p ( R ∗ 1 ). By definition of p ∗ 1 ( R 1 ) and p ∗ 2 ( R 1 ), p ( R 1 ) ≥ p ∗ 1 ( R 1 ) + p ∗ 2 ( R 1 ). Again b ecause of trimming, the p rofit collected by run R 2 is no smaller than 1 3 γ of the larger profit left either in runs R ∗ 1 or R ∗ 2 after requests serviced by run R 1 ha v e b een remov ed. Thus, p ( R 2 ) ≥ max  1 3 γ  p ( R ∗ 1 ) − p ∗ 1 ( R 1 )  , 1 3 γ  p ( R ∗ 2 ) − p ∗ 2 ( R 1 )   3 As a consequen ce, the total v a lue of p rofit collected by R 1 and R 2 is: p ( R 1 ) + p ( R 2 ) ≥ p ( R 1 ) + max  1 3 γ  p ( R ∗ 1 ) − p ∗ 1 ( R 1 )  , 1 3 γ  p ( R ∗ 2 ) − p ∗ 2 ( R 1 )   ≥ p ( R 1 ) + 1 2  1 3 γ  p ( R ∗ 1 ) − p ∗ 1 ( R 1 )  + 1 3 γ  p ( R ∗ 2 ) − p ∗ 2 ( R 1 )   = p ( R 1 ) + 1 6 γ p ( R ∗ 1 ) − 1 6 γ p ∗ 1 ( R 1 ) + 1 6 γ p ( R ∗ 2 ) − 1 6 γ p ∗ 2 ( R 1 ) ≥ p ( R 1 ) + 1 6 γ  p ( R ∗ 1 ) + p ( R ∗ 2 )  − 1 6 γ  p 1 ( R 1 ) + p 2 ( R 1 )  ≥ 6 γ − 1 6 γ p ( R 1 ) + 1 6 γ  p ( R ∗ 1 ) + p ( R ∗ 2 )  ≥ 6 γ − 1 18 γ 2 p ( R ∗ 1 ) + 1 6 γ  p ( R ∗ 1 ) + p ( R ∗ 2 )  ≥ 6 γ − 1 36 γ 2 ( p ( R ∗ 1 ) + p ( R ∗ 2 )) + 1 6 γ  p ( R ∗ 1 ) + p ( R ∗ 2 )  ≥ 12 γ − 1 36 γ 2  p ( R ∗ 1 ) + p ( R ∗ 2 )  ✷ F or the 2-v ehicle p roblem on a tree, w here γ = 1, algorithm 2VEHICLE obtains at lea st 11 36 of total p ossible profit. Note th at this greedy m ulti-pass algorithm is similar to the one used for m ultiple-path orien teering in [2]. Using an analysis tec hnique f r om there, w e could achiev e a (3 γ + 1)-approximat ion to the Multiv ehicle Routing Problem. How ev er, w ith the more adv anced analysis giv en, we ac hiev e a b etter appr o ximation for 2 as well as other small num b ers of v ehicles. Our tec hnique can b e extended to 3 v ehicles. F or a tree, the app ro ximation ratio is 243 / 71 , as sho wn b elo w. p ( R 1 ) + p ( R 2 ) + p ( R 3 ) ≥ p ( R 1 ) + p ( R 2 ) + max  1 3  p ( R ∗ 1 ) − p ∗ 1 ( R 1 ) − p ∗ 1 ( R 2 )  , 1 3  p ( R ∗ 2 ) − p ∗ 2 ( R 1 ) − p ∗ 2 ( R 2 )  , 1 3  p ( R ∗ 3 ) − p ∗ 3 ( R 1 ) − p ∗ 3 ( R 2 )   ≥ p ( R 1 ) + p ( R 2 ) + 1 9  p ( R ∗ 1 ) − p ∗ 1 ( R 1 ) − p ∗ 1 ( R 2 )  + 1 9  p ( R ∗ 2 ) − p ∗ 2 ( R 1 ) − p ∗ 2 ( R 2 )  + 1 9  p ( R ∗ 3 ) − p ∗ 3 ( R 1 ) − p ∗ 3 ( R 2 )  = p ( R 1 ) + p ( R 2 ) + 1 9  p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 )  − 1 9  p ∗ 1 ( R 1 ) + p ∗ 2 ( R 1 ) + p ∗ 3 ( R 1 )  − 1 9  p ∗ 1 ( R 2 ) + p ∗ 2 ( R 2 ) + p ∗ 3 ( R 2 )  ≥ 8 9 p ( R 1 ) + 8 9 p ( R 2 ) + 1 9  p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 )  ≥ 8 9  11 36 ( p ( R ∗ 1 ) + p ( R ∗ 2 ))  + 1 9  p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 )  ≥ 8 9  22 108 ( p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 ))  + 1 9  p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 )  ≥ 71 243  p ( R ∗ 1 ) + p ( R ∗ 2 ) + p ( R ∗ 3 )  Our tec hnique logically extends to k vehicles on a tree. Let R ∗ 1 , R ∗ 2 , . . . R ∗ k b e the k d isjoin t optimal runs in d ecreasing order of p rofit. Let R 1 , R 2 , . . . R k b e the k runs pro d uced b y our algorithm. Using the same structure used for k = 2 an d k = 3, k X i =1 p ( R i ) ≥ k − 1 X i =1 p ( R i ) + ma x 1 ≤ i ≤ k ( 1 3 p ( R ∗ i ) − k − 1 X j =1 p ∗ i ( R j ) !) ≥ 3 k − 1 3 k k − 1 X i =1 p ( R i ) + 1 3 k k X i =1 p ( R ∗ i ) 4 W e can sub stitute the approximat ion f or the first k − 1 run s in to the first term of the b ound giv en ab o v e. Then, that term will still only giv e an appro ximation as a ratio of P k − 1 i =1 p ( R ∗ i ). Because w e know that the pr ofi t of R ∗ k is no greater than the profit of any of the ru ns in the sum , w e can b ound the approximat ion in terms of a ratio of P k i =1 p ( R ∗ i ) with an additional factor of ( k − 1) /k . Th us, for trees we find at least a P ( k ) f r action of optimal profit, where P ( k ) is defin ed as follo ws: P (1) ≥ 1 3 P ( k ) ≥  3 k 2 − 4 k + 1 3 k 2  P ( k − 1) + 1 3 k F or general metric graph s w ith a 3 γ -appr o ximation to th e T ra v eling Repairman Problem, we can generalize the analysis and the r ecurrence giv en to in clude a factor of γ . V ery similar analysis sho ws that w e find a related P γ ( k ) fr action of profit, defin in g P γ ( k ) as follo ws: P γ (1) ≥ 1 3 γ P γ ( k ) ≥  3 γ k 2 − (3 γ + 1 ) k + 1 3 γ k 2  P γ ( k − 1) + 1 3 γ k In T able 1 we list v alues of P γ ( k ) for b oth tree and graph versions for sev eral v alues of k . Note that the quality of the ap p ro ximation degrades very slo wly . A simp le pr o of by induction establishes that our approxi mation factor is alw a ys b etter than the (3 γ + 1) -approximati on we could h a v e ac hiev ed using the style of analysis from [2 ]. k γ = 1 γ = 2 + ǫ when ǫ = 1 2 11 36 ≈ 0 . 3056 35 324 ≈ 0 . 1080 3 71 243 ≈ 0 . 2922 698 6561 ≈ 0 . 1064 4 1105 3888 ≈ 0 . 2842 16589 157464 ≈ 0 . 1054 8 31871 118098 ≈ 0 . 2699 16016004 79 15496819 560 ≈ 0 . 1034 16 52434460 7599 20083878 14976 ≈ 0 . 2611 27557774 42862301661 27017034 353459841780 ≈ 0 . 1020 T able 1: V alues of P γ ( k ) for selected k an d γ . Although almost 50% b etter p er f ormance can b e gained with small ǫ , we let ǫ = 1 for simplicit y of presenta tion. 4 V ehicle Routing on a T ree When a S ingle V ehicle S uffices W e n o w consider an appro ximation algorithm for a sp ecial case of a minimum v ehicle routin g problem with unit-time windows. W e call this problem the M inimum V ehicle O P T = 1 Pr oblem . The in put to th is problem is identica l to the T rav eling Repairman Problem defined in Section 2. W e also assume th at only a single rep airm an is needed to service all requests. Belo w we defin e an algorithm called S INGLE-REP AIR that find s n o more than 6 indep endent runs that w ill service all requests, for the case of tree metrics. Should our algorithm not p ro duce 6 run s that s er v ice all requests, then we w ill kno w that no single vehicl e service tour exists. W e b egin SINGLE-REP AIR by making divisions in time sp aced one unit apart, s tarting at time 0. Let a p erio d b e the time interv al from one division u p to but not in cluding th e next division. Note that this step defi n es p erio ds to b e t wice as long as that giv en in the algo rithm describ ed in S ection 2. W e num b er the p erio d s in order of increasing time, starting with the num b er 0. If needed, we p erturb the time windows b y some negligible amount so that no time windo w b egins 5 exactly on a d ivision. Thus, ev ery time win do w will int ersect exactly t wo p erio ds . W e expand eac h time windo w s o that it fills b oth of the p erio ds it int ersects, doub lin g its length. W e partition the time w indo ws into t w o sets: I f the fir s t of the t wo p erio d s a time window fills is even, we put the windo w in set E . Otherwise, we put the window in set O . W e ru n the r ep airman algorithm on trimmed wind o ws from [6] on set E to fin d a sh ortest run R E servicing the maximum num b er of requests in that set. Then, we ru n the same algorithm on the windo ws in set O resulting in a run R O . Th is algorithms gives optimal r uns for trim m ed w indo ws on a tree; thus, R E and R O are optimal run s on their resp ectiv e sets. Since we kno w that a single v ehicle can service the requests of all of the unexpanded win do ws, R E and R O m ust service the requests of all of the w indo ws in their sets. In ord er to con v ert R E and R O in to r uns on u nexpanded windows, we mak e t w o additional copies of eac h. F or b oth sets of r uns, we mov e one copy bac k in time by 1 time unit and one cop y forw ard in time by 1 time unit. Theorem 4.1 Algorith m SINGLE- REP AIR finds a 6-appr oximation to the M inimum V ehicle O P T = 1 P r o blem when the underlying gr a ph is a tr e e. Pr o of: Let u s consider the case for p ath R E . If R E services a request of an expanded time win do w at time t , then the corresp ondin g unexpanded time window either con tains t , p recedes t , or follo ws t . If the original time windo w con tains t , then R E services its requ est. If the time window p recedes t , then the cop y of R E starting 1 time unit earlier m ust service its requ est. If the time window follo ws t , then the copy of R E starting 1 time un it later m ust service its request. Because the argumen t is identi cal for R O , our algorithm fin ds 6 p aths w hic h service all service requests. ✷ References [1] C. Bazgan, R. Hassin, and J . Monnot. Appro ximation algorithms for some vehicle routing problems. Discr ete Appl. Math. , 146(1):27– 42, 2005. [2] A. Blum, S . Ch a wla, D. R. K arger, T. Lane, A. Mey erson, and M. Mink off. Appr o ximation algorithms for orien teering and discounted-rew ard TS P. SIAM J. Comput. , 37(2):653–6 70, 2007. [3] C. Chekuri, N. Korula, and M. P ´ al. Improv ed algorithms for orientee ring and related p r oblems. In P r o c. 19th ACM-SIAM Symp. on Discr ete Algorith ms , pages 661 –670, Philadelphia, P A, USA, 2008. So ciet y for Indu strial and App lied Mathematics. [4] G. B. Dan tzig and J . H. Ramser. The tr u c k disp atc hing problem. Management Scienc e , 6:80–9 1, 1959. [5] J. F ak c haro enph ol, C. Harrelson, and S. Rao. The k-trav eling rep airman problem. In Pr o c. 14th ACM-SIAM Symp. on Discr ete Algorithm s , pages 655–66 4, 2003. [6] G. N. F r ederic kson and B. Wittman. Approxima tion algorithms for the trav eling rep airman and sp eeding deliveryman pr oblems with unit-time wind o ws. In APP R OX-RANDOM , vol ume 4627 of LNCS , pages 119–1 33. Sp ringer, 2007. [7] G. N. F red er ickson and B. Wittman. Appro ximation algorithms for the tra v eling re- pairman and sp eeding d eliveryman problems. Journal version, in sub mission, 2009, http://a rxiv.org /abs/0905.4444 . 6 [8] H. Hashimoto, T . Ib araki, S. Imahori, and M. Y agiura. The vehicle r outing p roblem with flexible time windows and tra v eling times. Discr ete Appl. Math. , 154(16 ):2271–2 290, 2006 . [9] Y. Karuno and H. Nagamo c hi. An appr o ximabilit y result of the m ulti-v ehicle scheduling problem on a p ath with r elease and hand lin g times. The or. Comput. Sci . , 312(2 -3):267–2 80, 2004. [10] V. Nagara jan and R. Ra vi. Minimum vehicl e routing with a common deadline. In AP P R OX- RANDOM , v olume 4110 of LNCS , pages 212–223 , 2006. 7