Minimum-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

Minimum-w eight double-tr ee shortcut ting for Metric TSP : Bounding the approximation ratio ✩ Vladimir Deinek o a,b , Alexander Tiskin a,c a Centr e for Discr ete Mathematics and Its Applic ations (DIMAP), University of Warwick, Coventry, CV4 7AL, UK. b Warwick Business Scho ol, Uni versity of Warwick, Coventry CV4 7AL, UK. c Dep artment of Computer Scienc e, University of Warwick, Coventry CV4 7AL, UK. Abstract The Met ric T ra v eling Salesman Problem (TSP) is a cla ssical NP-hard op- timizatio n problem. The double-tree sh ortcutting met ho d for Me tric TSP yields an exp onen tially-sized space of TSP tours, eac h of whic h approxima tes the optimal solution within at most a factor of 2. W e consider the p roblem of fin d ing among these tour s the one that gives the closest a ppr o ximation, i.e. the minimum-weight double-tr e e shortcutting . Previously , we g a ve an ef- ficien t algorithm for this problem, and ca rried out its exp erimenta l analysis. In this pap er, w e address the related question of the w orst-case appro xi- mation r atio f or the minimum-w eigh t double-tree sh ortcutting metho d. In particular, we giv e lo w er b oun ds on the approximat ion r atio in some sp e- cific metric spaces: the ratio of 2 in th e discrete shortest path metric, 1.622 in the planar E uclidean metric, and 1.666 in the p lanar Minko wski metric. The first of these lo w er b ound s is tight ; we co njecture that the other t w o b ound s are also tigh t, and in particular that the minim um-w eigh t double- tree metho d provides a 1 . 62 2-appro ximation for p lanar Euclidean TSP . 1. Intro duction The Metric T rav elling Salesman Problem (TSP) is a classical com binato- rial optimization p roblem. W e represent a set of n p oint s in a metric space b y a complete w eigh ted graph on n no des, where the w eigh t of an edge is defined b y the distance b et w een the corresp ondin g p oin ts. The ob jectiv e of Metric TSP is to find in this graph a min im um-we igh t Hamiltonian cycle ✩ Researc h supp orted by the Centre for Discrete Mathematics and Its Applications (DIMAP), Universit y of W arwick. Pr eprint submitte d to Elsevier Octob er 30, 2018 (equiv alentl y , a min im um-w eigh t tour visiting ev ery no de at least once). The most common example of Metric TSP is the planar Eu clidean TSP , where the p oints lie in the tw o-dimensional Euclidean p lane, and the d istances are measured according to the Euclidean metric. Metric TS P , even restricted t o p lanar Euclidean TSP , is well-kno wn to b e NP-hard [10]. Metric TSP is also kno wn to b e NP-hard to appro ximate to within a ratio 1 . 0045 6, but p olynomial-time approximable to within a ratio 1 . 5. Fixed-dimension Euclidean TSP is kno wn to ha v e a PT AS (i.e. a family of algo rithms with appro ximation r atio arb itrarily clo se to 1) [1]; this generalises to any metric defined by a fixed-dimension Mink o wski vec tor norm. Tw o simple metho ds, double-tree shortcutting [12] and C hristofides’ [4, 13], allo w one to app ro ximate the solution of Metric TSP within a facto r of 2 and 1 . 5, resp ectiv ely . Bot h these metho ds belong to the class of tour- c onstructing heuristics , i.e. “heur istics that incrementa lly constru ct a tour and stop as so on as a v alid tour is created” [7]. In b oth metho ds, we build an Eulerian graph on the giv en p oint set, select an Euler tour of the graph, and then p erform shortcutting on this tour b y remo ving rep eated no d es, unt il all n o de rep etitions are remo v ed. In general, it is not p rescrib ed wh ic h one of sev eral occurr ences of a p articular no d e to r emo v e. Therefore, the metho ds yield an exp onen tially-sized sp ace of TS P tours (shortcuttings of a sp ecific E u ler tour in a sp ecific Eulerian graph), eac h of wh ic h approximat es the optimal solution with in at most a factor of 2 (resp ectiv ely , 1 . 5). The t wo metho ds differ in the wa y the initial w eigh ted Eu lerian grap h is constructed. Both start by fi nding the graph’s minim um-weig ht sp anning tree (MST). Th e doub le-tree metho d then doubles every edge in the MST , while the Chr istofides metho d adds to the MST a minimum-w eigh t matc hing built on the set of o dd-degree no des. The weig ht of th e resulting Euler tour is h igher than the optimal TSP tour at most b y a factor of 2 (resp ectiv ely , 1 . 5), and the su bsequent shortcutting can only decrease the tour we igh t. While any tour obtained by shortcutting of the original Euler tour ap - pro ximates the optimal solution within at most a factor of 2 (resp ectiv ely , 1 . 5), clearly , it is still desirable to find the shortcutting that give s the clos- est appro ximation. Giv en an Eulerian graph on a set of p oin ts, w e will consider its minimum-weight shortcutting across all sh ortcuttings of all p os- sible Euler tours of th e graph. W e sh all corr esp ondingly sp eak ab out the minimum-weight double-tr e e and the minimum-weight Christofides metho ds. Unfortunately , for the general Metric T SP (i.e. an arbitrary complete w eigh ted graph with the triangle inequalit y), the corresp onding double-tree and Christofid es minimum-w eigh t shortcutting problems are b oth NP-hard. 2 1 ǫ 1 (a) Minim um spanning tree (b) D epth-first dou b le-tree tour (c) Absolute minim um- w eight tour Figure 1: The depth-first doub le-tree method : a lo w er-b ound instance The min imum-w eigh t double-tree s h ortcutting pr oblem w as also b eliev ed f or a long time to b e NP-hard for planar Euclidean TSP , until a p olynomial- time algo rithm was given by Burk ard et al. [3]. In [6], we ga v e an improv ed algorithm ru nning in time O (4 d n 2 ), wh ere d is th e maxim um n o de degree in the ro oted minim um spanning tree (e.g. in the non-degenerate planar Euclidean case, d ≤ 4). In contrast , the Christofides v ersion of the problem remains NP-hard ev en for planar Euclidean TS P [11]. A natur al question ab out the prop er ties of the tw o appr o ximation meth- o ds and their v arian ts is whether the app ro ximation ratios 2 and 1 . 5 are tigh t, i.e. whether there is a p r oblem in stance where the appro ximate so- lution h as app ro ximation ratio 2 (resp ectiv ely , 1 . 5), or a family of problem instances where the appr o ximate solutions approac h these ratios arb itrarily closely . F or the minimum-w eigh t double-tree metho d, the answ er to this qu es- tion is unkno wn, as observ ed e.g. in [9]. The only existing low er b ounds f or the double-tree m etho d apply to a shortcutting that is p er f ormed in some sub optimal, easily computable order. An example of suc h an order is depth- first tree tra v ersal; we shall call the resu lting metho d depth-first double-tr e e shortcutting . A tigh t lo w er b ound for this metho d is giv en b y the stan- dard Euclidean lo we r-b ound construction sh o wn in Figure 1, whichadapted from [8]. Figure 1a sho ws an instance p oin t set and the (uniqu e) m in i- m um spanning tree. W e assume that ǫ = o (1); for example, w e can tak e ǫ = 1 /n . Th e v ertical size of the instance set is 1, and the horizon tal size is  1 + o (1)  n . The weigh t of th e unique MST is  2 + o (1)  n ; the doub le-tree w eigh t is  4 + o (1)  n . Th e double tree u ndergo es no significant sh ortcut- ting, and the resulting tour (Figure 1b) still has weigh t  4 + o (1)  n . The absolute m inim um-wei ght tour (Figure 1c) has weigh t  2 + o (1)  n , therefore the approximat ion r atio on the giv en in stance set is 2. F or the minim um-weig ht Chr istofides algorithm, a tigh t lo w er b ound is giv en by the standard Eu clidean lo w er-b ound construction shown in Fig- ure 2 , wh ic h is adapted from [5] and uses the same con v en tions as Figure 1. 3 b b b b b 1 ǫ 1 (a) Minim um spanning tree b b b b b (b) Minim um-weigh t Christofides tour b b b b b (c) Absolute minim um- w eight tour Figure 2: The minimum-w eigh t Christofid es metho d : a lo wer-b ound in- stance The minimum spann ing tree has exactly t w o o dd -degree nod es, therefore the add itional m atching consists of a single edge. The resu lting Eulerian graph (Fig ure 2b) is already a Hamilt onian cycle, hence no s hortcutting is r equired. The w eigh t of the cycle is  3 + o (1)  n . As b efore, the abso- lute min im um-we igh t tour (Figure 2c) has w eigh t  2 + o (1)  n , therefore the appro ximation ratio on the giv en instance sets is 1 . 5. In the rest of this pap er, we address the qu estion of th e wo rst-case ap - pro ximation ratio for th e minim um-weig ht double-tree shortcutting m etho d in some sp ecific metric spaces. In particular, we give a lo w er b ound on the approximati on ratio in the discrete sh ortest path metric 1 ; this b ound is tigh t, and can b e r egarded as a lo w er b ound f or a generic metric space. W e also giv e the fir st non-trivial lo wer b ound for the planar Euclidean an d planar Mink o wski metrics. 2. The low er b ounds 2.1. The discr ete shortest p ath metric The w orst-case appro ximation r atio of the double-tree and Christofides metho ds can clearly b e dep endent on the metric in w hic h the TS P prob lem is defi ned. In the Introd uction, w e describ ed a tigh t lo we r b ound of 2 on the w orst-case appro ximation ratio in th e planar Euclidean metric, b oth for the depth-first version of the doub le-tree metho d and for th e minimum-w eigh t Christofides metho d. In contrast, n o non-trivial lo w er b ounds h a v e b een kno wn, to our knowle dge, for the min im um-we igh t double-tree metho d in an y metric. A tigh t b ound in the Eu clidean m etric seems difficult to obtain; ho w ev er, it can b e established that the upp er b ound of 2 is tigh t in some non-Euclidean metrics, an d therefore is tigh t for the generic Metric TSP. 1 The same result has b een obtained in d ep en dently by Bil` o at al. [2]. 4 b b b b b b b b b b b b b b b b 1 1 + ǫ (a) Minim um spanning tree b b b b b b b b b b b b b b b b (b) Minim um-weigh t double-tree tour b b b b b b b b b b b b b b b b (c) Absolute minim um- w eight tour Figure 3: The minim um-we igh t d ouble-tree metho d: a lo w er-b ound instance in the discrete sh ortest path metric Giv en a we igh ted un d irected graph , consider the discr ete shortest p ath metric on its nod e set. The distance b et w een tw o no des in this metric is defined as the w eigh t of the sh ortest p ath connecting them in the graph. Let n b e a p o w er of 2. L et T n b e a ro oted tree on n no des, wher e the ro ot has a single child, which branc hes off in to a complete binary tree w ith n/ 2 lea v es. W e construct an instance graph on 2 n no d es as follo ws. First, we create t w o copies of the tree T n , k eeping trac k of corresp ondin g pairs of n o des (i.e. pairs of n o des whic h are copies of the same n o de in T n ). W e th en giv e all the edges in eac h tree w eigh t 1, and connect the ro ots of the t w o trees by a r o ot e dge of weigh t 1. Finally , w e connect ev ery pair of corresp onding non-ro ot no des in b oth trees by a cr oss-e dge of we ight 1 + ǫ . W e assume that ǫ = o (1); for example, w e can tak e ǫ = 1 /n . The instance graph corresp onding to n = 8 is shown in Figure 3a, wh ere ed ges of w eigh t 1 and 1 + ǫ are represented, resp ectiv ely , by solid and dotted lines. The uniqu e MST consists of b oth copies of T n plus the ro ot edge, and has w eigh t  2 − o (1 )  n ; the double-tree w eigh t is  4 − o (1 )  n . Note th at for an y tw o no d es a , b w ith in the same cop y of T n , the distance b et w een a an d b is equal to the weigh t of the path connecting these n o des in the tree. Hence, a sh ortcutting f rom a, b, c to a, c can redu ce the tour weig ht, only if a and c b elong to differen t copies of T n . Also note that an y double-tree Euler tour of T n has w eigh t 2 n − 2. An y Hamiltonian cycle of the complete w eigh ted graph obtained b y shortcutting the d ouble-tree Euler tour will con tain a Hamiltonian path in a complete weig hte d subgraph ind uced by eac h cop y of T n . The weigh t of a su c h a Hamiltonian p ath can differ f rom the w eigh t of the d ouble-tree tour of T n b y at most the w eigh t of a single edge, whic h cannot exceed 2 log n = o ( n ). Therefore, the resulting Hamiltonian cycle still has weig ht  4 − o (1)  n . 5 b b b b b b b b b b b b b b b b b b b b b b b b b 1 b b b b b b b b b (a) Minim um spanning tree b b b b b b b b b b b b b b b b b b b b b b b b b (b) Minim um-weigh t double-tree tour b b b b b b b b b b b b b b b b b b b b b b b b b (c) Absolute minim um- w eight tour Figure 4: The minim um-we igh t d ouble-tree metho d: a lo w er-b ound instance in the Euclidean and Minko wski metrics The min imum-w eigh t double-tree tour for our example is sho wn in Fig- ure 3b, where straigh t edges h a v e w eigh ts 1 and 1 + ǫ , and cur v ed edges ha v e in teger w eigh ts greater than 1. An edge’s curv ature indicates the la y- out of the shortest path along whic h the edge we ight is measured. The absolute min im um-we igh t tour has we ight  2 + o (1)  n , and consists of the ro ot edge and all the cross-edges, link ed together by edge-disjoin t paths in the tw o trees. The absolute minim um-weigh t tour f or ou r example is sho wn in Figure 3c, u sing the same graphic con ve ntio ns as in Figur e 3b. The ap- pro ximation ratio of the minim um-wei ght double-tree metho d on the give n instance set is 4 / 2 = 2, which matc hes the generic u p p er b ound 2 . 2.2. Euclide an and M i nkowski metrics Compared with the ab o ve constru ction f or the discrete s hortest path metric, it app ears to b e m uc h more d ifficult to obtain a tight b ound in planar Euclidean-t yp e m etrics. W e d escrib e a construction that provides the first non-trivial lo w er b ound on the approxima tion r atio of the minim um-weigh t double-tree metho d in the planar Euclidean and Minko wski metrics. The prop osed construction consists of 6 n + 1 p oints, and is sho wn in Figure 4a for n = 4. T he instance p oin t set consists of sev en p oints f orming a symm etric three-w a y cen tral star of arbitrary constan t size, and six r o ws of p oin ts extend ing fr om the star’s ends in three symmetric directions in s teps of length 1. Figure 4a sho ws the (un iqu e) minimum spanning tree, w hic h has w eigh t  6 + o (1)  n . Figure 4b sho ws the min imum-w eigh t d ouble-tree 2 Man y v ariations on the describ ed construct ion are p ossible. W e hav e chosen a v ariant that is easy to visualise. 6 tour, wh ic h has weigh t  8 + √ 3 + o (1)  n . Figure 4c shows the absolute minim um-weig ht tour, whic h has w eigh t  6 + o (1)  n . The appr o ximation ratio of the minimum-w eigh t double-tree metho d on the giv en instance set is (8 + √ 3) / 6 ≈ 1 . 62 2. There remains a su bstan tial gap b et w een this lo w er b ound and the generic upp er b oun d of 2, wh ic h is also th e b est kno wn upp er b ound in the planar Eu clidean metric. The same constru ction pro vides a somewhat stronger lo w er b oun d in a metric defi n ed b y the hexa gonal norm — a Mink o wski v ecto r norm with the u nit disc in the shap e of a regular hexagon (see Figure 4a). In th is metric, the distance b et w een t wo p oin ts is measured along a p olygonal path comp osed from segments p arallel to the edges of the u nit d isc. The weigh ts of the minimum spannin g tree (Figure 4a) and of the absolute minim um- w eigh t tour (Figure 4c) on the ab o v e instance set r emain asymptotically unc hanged in the n ew metric. Ho w ev er, the wei ght of the minim um-weig ht double-tree tour (Figure 4b) increases to  10 + o (1)  n . Therefore, the lo w er b ound in the hexagonal m etric is 10 / 6 ≈ 1 . 666 . 3. C onclusions In the pr evious section, w e presen ted lo w er b ounds on the minim um- w eigh t double-tree m etho d . W e ha v e sho wn th at the trivial up p er b oun d of 2 is tight in at least some metrics (in particular, the d iscrete sh ortest path metric). Ho we ve r, in the imp ortan t cases of the Euclidean and Minko w s ki metrics, a substantia l gap remains b et w een our low er b ounds of 1.622 (re- sp ectiv ely , 1.666) and the trivial u pp er b oun d of 2. Considering the appar- en t difficult y of impro ving on th ese lo w er b ound s, and the goo d appro xi- mation b ehavio ur of the minimum-w eigh t double-tree algorithm on t ypical Euclidean TS P instances [6], w e conjecture that these lo w er b oun ds are tigh t, and that th e min imum-w eigh t doub le-tree metho d p r o vides a 1 . 622- appro ximation for planar Euclidean TSP . References [1] S. Arora. Po lynomial-time approximat ion sc hemes for Euclidean TSP and other geomet ric p roblems. J ournal of the ACM , 45:753–782 , 1998. [2] D. Bil` o, L. F orlizzi, and G. Proietti. Appro ximating the Metric TSP in linear time. 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