Fast minimum-weight double-tree shortcutting for Metric TSP: Is the best one good enough?
aa r X i v : . [ c s . D S ] J u l Fast minimum-weight double-tree shortcutting forMetric TSP: Is the best one good enough?
VLADIMIR DEINEKOWarwick Business School and DIMAP, University of WarwickCoventry CV4 7AL, UKandALEXANDER TISKINDepartment of Computer Science and DIMAP, University of WarwickCoventry CV4 7AL, UK
The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. Thedouble-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours,each of which approximates the optimal solution within at most a factor of 2. We considerthe problem of finding among these tours the one that gives the closest approximation, i.e. the minimum-weight double-tree shortcutting . Burkard et al. gave an algorithm for this problem, run-ning in time O ( n + 2 d n ) and memory O (2 d n ), where d is the maximum node degree in therooted minimum spanning tree. We give an improved algorithm for the case of small d (includingplanar Euclidean TSP, where d ≤ O (4 d n ) and memory O (4 d n ). This improve-ment allows one to solve the problem on much larger instances than previously attempted. Ourcomputational experiments suggest that in terms of the time-quality tradeoff, the minimum-weightdouble-tree shortcutting method provides one of the best known tour-constructing heuristics.Categories and Subject Descriptors: E.1 [ DATA STRUCTURES ]: Graphs and networks; F.2.2[
ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY ]: Nonnumerical Al-gorithms and Problems—
Computations on discrete structures ; G.2.2 [
DISCRETE MATHE-MATICS ]: Graph Theory—
Graph algorithms
General Terms: Algorithms, Experimentation, Performance, TheoryAdditional Key Words and Phrases: Approximation algorithms, Metric TSP, double-tree short-cutting
1. INTRODUCTION
The Metric Travelling Salesman Problem (TSP) is a classical combinatorial opti-mization problem. We represent a set of n points in a metric space by a completeweighted graph on n nodes, where the weight of an edge is defined by the distancebetween the corresponding points. The objective of Metric TSP is to find in this Research supported by the Centre for Discrete Mathematics and Its Applications (DIMAP), Uni-versity of Warwick, and by EPSRC fund EP/F017871.Permission to make digital/hard copy of all or part of this material without fee for personalor classroom use provided that the copies are not made or distributed for profit or commercialadvantage, the ACM copyright/server notice, the title of the publication, and its date appear, andnotice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish,to post on servers, or to redistribute to lists requires prior specific permission and/or a fee.c (cid:13)
ACM Journal Name, Vol. V, No. N, Month 20YY, Pages 1–0 ?? . · V. Deineko and A. Tiskin graph a minimum-weight Hamiltonian cycle (equivalently, a minimum-weight tourvisiting every node at least once). The most common example of Metric TSP isthe planar Euclidean TSP, where the points lie in the two-dimensional Euclideanplane, and the distances are measured according to the Euclidean metric.Metric TSP, even restricted to planar Euclidean TSP, is well-known to be NP-hard [Papadimitriou 1977]. Metric TSP is also known to be NP-hard to approximateto within a ratio 1 . . double-tree method [Rosenkrantz et al. 1977] and the Christofides method [Christofides 1976; Serdyukov 1978], allow one to approximatethe solution of Metric TSP within a factor of 2 and 1 .
5, respectively. Both methodsbelong to the class of tour-constructing heuristics , i.e. “heuristics that incremen-tally construct a tour and stop as soon as a valid tour is created” [Johnson andMcGeoch 2002]. In both methods, we build an Eulerian graph on the given pointset, select an Euler tour of the graph, and then perform shortcutting on this tourby removing repeated nodes, until all node repetitions are removed. In general, itis not prescribed which one of several occurrences of a particular node to remove.Therefore, the methods yield an exponentially-sized space of TSP tours (shortcut-tings of a specific Euler tour in a specific Eulerian graph), each approximating theoptimal solution within a factor of 2 (respectively, 1 . . minimum-weight shortcutting across all shortcuttingsof all possible Euler tours of the graph. We shall correspondingly speak about theminimum-weight double-tree and the minimum-weight Christofides methods.Unfortunately, for general Metric TSP, both the double-tree and Christofidesminimum-weight shortcutting problems are NP-hard. Consider an instance of theHamiltonian cycle problem on an unweighted graph; this can be regarded as aninstance of Metric TSP with weights 1 and 2. Add an extra node connected toall the original nodes by edges of weight 1, and take the newly added edges as theMST. It is easy to see that the resulting minimum-weight double-tree shortcuttingproblem is equivalent to the original Hamiltonian cycle problem. The minimum-weight double-tree shortcutting problem was believed for a long time to be NP-hardeven for planar Euclidean TSP, until a polynomial-time algorithm was given byBurkard et al. [1998]. This is the algorithm we improve upon in the current paper.In contrast, the minimum-weight Christofides shortcutting problem remains NP- ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · hard even for planar Euclidean TSP [Papadimitriou and Vazirani 1984].In the rest of this paper, we will mainly deal with the rooted MST , which isobtained from the MST by selecting an arbitrary node as the root . In the rootedMST, the terms parent , child , ancestor , descendant , sibling , leaf all have theirstandard meaning. Let d denote the maximum number of children per node inthe rooted MST. Note that in the Euclidean plane, the maximum degree of anunrooted MST is at most 6. Moreover, a node can have degree equal to 6, only if itis surrounded by six equidistant nodes forming a regular hexagon; we can excludethis degenerate case from consideration by a slight perturbation of the input points.This leaves us with an unrooted MST of maximum degree 5. By choosing a nodeof degree less than 5 as the root, we obtain a rooted MST with d ≤ O ( n + 2 d n ) and memory O (2 d n ). In this paper, we give an improved algorithm for the case of small d ,running in time O (4 d n ) and memory O (4 d n ). In the planar Euclidean case, bothabove algorithms run in polynomial time and memory.We then describe our implementation of the new algorithm, which incorporatesa couple of additional heuristic improvements designed to speed up the algorithmand to increase its approximation quality. Computational experiments show thatthe approximation quality and running time of our implementation are among thebest known tour-constructing heuristics.A preliminary version of this paper appeared as [Deineko and Tiskin 2007].
2. THE ALGORITHM2.1 Preliminaries
Let G be a weighted graph representing the Metric TSP problem on n points. Thedouble-tree method consists of the following stages:—construct the minimum spanning tree of G ;—duplicate every edge of the tree, obtaining an n -node Eulerian graph;—select an Euler tour of the double-tree graph;—reduce the Euler tour to a Hamiltonian cycle by repeated shortcutting , i.e. re-placing a node sequence a, b, c by a, c , as long as node b appears elsewhere in thecurrent tour.We say that a Hamiltonian cycle conforms to the doubled spanning tree, if it canbe obtained from that tree by shortcutting one of its Euler tours. We also extendthis definition to paths, saying that a path conforms to the tree, if it is a subpathof a conforming Hamiltonian cycle.In our minimum-weight double-tree shortcutting algorithm, we refine the bottom-up dynamic programming approach of [Burkard et al. 1998]. Initially, we select anarbitrary node r as the root of the tree. For a node u , we denote by C ( u ) the setof all children of u , and by T ( u ) the node set of the maximal subtree rooted at u , Note that Burkard et al. [Burkard et al. 1998] also give an O (2 d n ) algorithm for a more gen-eral TSP-type problem, where the set of admissible tours is restricted by a given PQ-tree. Ouralgorithm does not improve on the algorithm of [Burkard et al. 1998] for this more general problem. ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin i.e. the set of all descendants of u (including u itself). For a set of siblings U , wedenote by T ( U ) the (disjoint) union of all subtrees T ( u ), u ∈ U . When U is empty, T ( U ) is also empty.The characteristic property of a conforming Hamiltonian cycle is as follows: forevery node u , the cycle must contain all nodes of T ( u ) consecutively in some order.For an arbitrary node set S , we will say that a path through the graph sweeps S , if it visits all nodes of S consecutively in some order. In this terminology, aconforming Hamiltonian cycle must, for every node u , contain a subpath sweepingthe subtree T ( u ).In the rest of this section, we denote the metric distance between u and v by d ( u, v ). We use the symbol ⊎ to denote disjoint set union. For brevity, given a set A and an element a , we write A ⊎ a instead of A ⊎ { a } , and A \ a instead of A \ { a } . The algorithm proceeds by computing minimum-weight sweeping paths in progres-sively increasing subtrees, beginning with the leaves and finishing with the wholetree T ( r ). A similar approach is adopted in [Burkard et al. 1998], where in eachsubtree, all-pairs minimum-weight sweeping paths are computed. In contrast, ouralgorithm only computes single-source minimum-weight sweeping paths originatingat the subtree’s root. This leads to substantial savings in time and memory.A non-root node v ∈ C ( u ) is active , if its subtree T ( v ) has already been processed,but its parent’s subtree T ( u ) has not yet been processed. In every stage of thealgorithm, we choose the current node u , so that all children of u (if any) areactive. We call T ( u ) the current subtree . Let V ⊆ C ( u ), a ∈ T ( V ). We denote by D uV ( a ) the weight of the shortest conforming path starting from u , sweeping thesubtree u ⊎ T ( V ), and finishing at a .Consider the current subtree T ( u ). Processing this subtree will yield the values D uV ( a ) for all V ⊆ C ( u ), a ∈ T ( V ). In order to process the subtree, we need thecorresponding values for all subtrees rooted at the children of u . More precisely, weneed the values D vW ( a ) for every child v ∈ C ( u ), every subset W ⊆ C ( v ), and everydestination node a ∈ T ( W ). We do not need any explicit information on subtreesrooted at grandchildren and lower descendants of u .Given the current subtree T ( u ), the values D uV ( a ) are computed inductively forall sets V of children of u . The induction is on the size of the set V . The base ofthe induction is trivial: no values D uV ( a ) exist when V = ∅ .In the inductive step, given a set V ⊆ C ( u ), we compute the values D uV ⊎ v ( a ) forall v ∈ C ( u ) \ V , a ∈ T ( v ), as follows. By the inductive hypothesis, we have thevalues D uV ( a ) for all a ∈ T ( V ). The main part of the inductive step consists incomputing a set of auxiliary values D uV,W ( v ), for all subsets W ⊆ C ( v ). Every suchvalue represents the weight of the shortest conforming path starting from node u ,sweeping the subtree u ⊎ T ( V ), then sweeping the subtree T ( W ) ⊎ v , and finishingat node v . Suppose the path exits the subtree u ⊎ T ( V ) at node x and enters thesubtree T ( W ) ⊎ v at node y . We have ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · u b T (V ) b v b T (W ) bb x b y Fig. 1: Computation of D uV,W ( v ) u b T (V ) b v b T (v) b a (a) Case W = ∅ u b T (V ) b v bb T (v) b a (b) Case W = ∅ Fig. 2: Computation of D uV ⊎ v ( a ), a ∈ T ( v ) D uV,W ( v ) = d ( u, v ) if V = ∅ , W = ∅ min y ∈ T ( W ) (cid:2) d ( u, y ) + D vW ( y ) (cid:3) if V = ∅ , W = ∅ min x ∈ T ( V ) (cid:2) D uV ( x ) + d ( x, v ) (cid:3) if V = ∅ , W = ∅ min x ∈ T ( V ); y ∈ T ( W ) (cid:2) D uV ( x ) + d ( x, y ) + D vW ( y ) (cid:3) if V = ∅ , W = ∅ (1)(see Figure 1). The required values D vW ( y ) have been obtained previously, whileprocessing subtrees T ( v ) for the active nodes v ∈ C ( u ). Note that the computedauxiliary values include D uV ⊎ v ( v ) = D uV,C ( v ) ( v ).Now we can compute the values D uV ⊎ v ( a ) for all a ∈ T ( v ) \ v = T ( C ( v )). A pathcorresponding to D uV ⊎ v ( a ) must sweep u ⊎ T ( V ), and then T ( v ), finishing at a .While in T ( v ), the path will first sweep a (possibly single-node) subtree v ⊎ T ( W ),finishing at v . Then, starting at v , the path will sweep the subtree v ⊎ T ( W ), where W = C ( v ) \ W , finishing at a . Considering every possible disjoint bipartitioning W ⊎ W = C ( v ), such that a ∈ T ( W ), we have D uV ⊎ v ( a ) = min W ⊎ W = C ( v ): a ∈ T ( W ) (cid:2) D uV,W ( v ) + D vW ( a ) (cid:3) (2)(see Figure 2).We now have the values D uV ⊎ v ( a ) for all a ∈ T ( v ). The computation (1)–(2) isrepeated for every node v ∈ C ( u ) \ V . The inductive step is now completed.The processing of subtree T ( u ) terminates when all possible choices of subset V and node v have been exhausted.Eventually, the root r of the tree becomes the current node, and we process thecomplete tree T ( r ). This establishes the values D rS ( a ) for all S ⊆ C ( r ), a ∈ T ( S ),which includes the values D rC ( r ) ( a ) for all a = r . The weight of the minimum-weight ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin u bb T (V n v1) v1 bb T (C(v1) n v2) v2 bb T (C(v2) n a) a b T (a)
Fig. 3: Computation of P uV ( a ), a ∈ T ( V ), k = 3 conforming Hamiltonian cycle can now be determined asmin a = r (cid:2) D rC ( r ) ( a ) + d ( a, r ) (cid:3) (3) Theorem 2.1.
The upsweep algorithm computes the weight of the minimum-weight tree shortcutting in time O (4 d n ) and space O (2 d n ) . Proof.
In computation (1), the total number of quadruples u, v, x, y is at most n (since for every pair x , y , the node u is determined uniquely as the lowestcommon ancestor of x , y , and the node v is determined uniquely as a child of u and an ancestor of y ). In computation (2), the total number of triples u, v, a is alsoat most n (since for every pair u , a , the node v is determined uniquely as a childof u and an ancestor of y ). For every such quadruple or triple, the computation isperformed at most 4 d times, corresponding to 2 d possible choices of each of V , W .The cost of computation (3) is negligible. Therefore, the total time complexity ofthe algorithm is O (4 d n ).Since our goal at this stage is just to compute the solution weight, at any givenmoment we only need to store the values D uV ( a ), where u is either an active node, orthe current node (i.e. the node for which these values are currently being computed).When u corresponds to an active node, the number of possible pairs u, a is at most n (since node u is determined uniquely as the root of the active subtree containing a ). When u corresponds to the current node, the number of possible pairs u, a isalso at most n (since node u is fixed). For every such pair, we need to keep at most2 d values, corresponding to 2 d possible choices of V . The remaining space costs arenegligible. Therefore, the total space complexity of the algorithm is O (2 d n ). In order to reconstruct the minimum-weight Hamiltonian cycle itself, we must keepall the auxiliary values D uV,W ( v ) obtained in the course of the upsweep computationfor every parent-child pair u, v . We solve recursively the following problem: givena node u , a set V ⊆ C ( u ), and a node a ∈ T ( V ), find the minimum-weight path P uV ( a ) starting from u , sweeping subtree u ⊎ T ( V ), and finishing at a . To computethe global minimum-weight Hamiltonian cycle, it is sufficient to determine the path P rC ( r ) ( a ), where r is the root of the tree, and a is the node for which the minimumin (3) is attained.For any u , V ⊆ C ( u ), a ∈ T ( V ), consider the (not necessarily conforming orminimum-weight) path u = v → v → v → · · · → v k = a , joining nodes u and ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · a in the tree (see Figure 3). The conforming minimum-weight path P uV ( a ) firstsweeps the subtree u ⊎ T ( V \ v ). After that, for every node v i , 0 < i < k , thepath P uV ( a ) sweeps the subtree v i ⊎ T ( C ( v i ) \ v i +1 ) as follows: first, it sweeps asubtree v i ⊎ T ( W i ), finishing at v i , and then, starting at v i , it sweeps the subtree v i ⊎ T ( W i ), for some disjoint bipartitioning W i ⊎ W i = C ( v i ) \ v i +1 . Finally, thepath P uV ( a ) sweeps the subtree T ( a ), finishing at a .The optimal choice of bipartitionings can be found as follows. We construct aweighted directed layered graph with a source vertex corresponding to node u = v ,a sink vertex corresponding to node v k = a , and k − v i , 0 < i < k . Each intermediate layerconsists of at most 2 d − vertices, representing all different disjoint bipartitioningsof the node set C ( v i ) \ v i +1 . The source and the sink vertices represent the trivialbipartitionings ∅ ⊎ ( V \ v ) = V \ v and C ( a ) ⊎ ∅ = C ( a ), respectively. Everyconsecutive pair of vertex layers (including the source and the sink vertices) arefully connected by forward arcs. In particular, the arc from a vertex representingthe bipartitioning X ⊎ X in layer i , to the vertex representing the bipartitioning Y ⊎ Y in layer i +1, is given the weight D v i X,Y ( v i +1 ). It is easy to see that an optimalchoice of bipartitioning corresponds to the minimum-weight path from the sourceto the sink in the layered graph. This minimum-weight path can be found by astandard dynamic programming algorithm (such as the Bellman–Ford algorithm,see e.g. [Cormen et al. 2001]) in time proportional to the number of arcs in thelayered graph.Let W ⊎ W , . . . , W k − ⊎ W k − now denote the k − k arcs of the corresponding source-to-sink shortest path deter-mine k edges (not necessarily consecutive) in the minimum-weight sweeping path P uV ( a ). These edges are shown in Figure 3 by dotted lines. It now remains to applythe downsweep algorithm recursively in each of the subtrees u ⊎ T ( V \ v ), v ⊎ T ( W ), v ⊎ T ( W ), v ⊎ T ( W ), v ⊎ T ( W ), . . . , v k − ⊎ T ( W k − ), v k − ⊎ T ( W k − ), T ( a ). Theorem 2.2.
Given the output and the necessary intermediate values of theupsweep algorithm, the downsweep algorithm computes the edges of the minimum-weight tree shortcutting in time and space O (4 d n ) . Proof.
The construction of the layered graph and the minimum-weight pathcomputation runs in time O (4 d k ), where k is the number of edges in the tree path u = v → v → v → · · · → v k = a in the current level of recursion. Since thetree paths in different recursion levels are edge-disjoint, the total number of edgesin these paths is at most n . Therefore, the time complexity of the downsweepalgorithm is O (4 d n ).By Theorem 2.1, the space complexity of the upsweep algorithm is O (2 d n ). Inaddition to the storage used internally by the upsweep algorithm, we also need tokeep all the values D uV,W ( v ). The number of possible pairs u, v is at most n (sincenode u is determined uniquely as the parent of v ). For every such pair, we need tokeep at most 4 d values, corresponding to 2 d possible choices of each of V , W . Theremaining space costs are negligible. Therefore, the total space complexity of thedownsweep algorithm is O (4 d n ). ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin
3. HEURISTIC IMPROVEMENTS
Despite the guaranteed approximation ratio of the double-tree shortcutting andChristofides methods, neither has performed well in previous computational experi-ments (see [Johnson and McGeoch 1997; Reinelt 1994]). However, to our knowledge,none of these experiments explored the minimum-weight double-tree shortcuttingapproach. Instead, the double-tree shortcutting was performed in some subopti-mal, easily computable order, such as a depth-first tree traversal. We shall call thismethod depth-first double-tree shortcutting .In particular, [Reinelt 1994] compares 37 tour-constructing heuristics, includingthe depth-first double-tree algorithm and the Christofides algorithm, on a set of24 geometric instances from the TSPLIB database [Reinelt 1991]. Although mostinstances in this experiment are quite small (2000 or fewer points), they still allowus to make some qualitative judgement about the approximation quality of differentheuristics. Depth-first double-tree shortcutting turns out to have the lowest qualityof all 37 heuristics, while the quality of the Christofides algorithm is somewhathigher, but still far from the top.Intuitively, it is clear that the reason for the poor approximation quality of thetwo algorithms may be in the wrong choice of the shortcutting order, especially con-sidering that the overall number of alternative choices is typically exponential. Thisobservation motivated us to implement the minimum-weight double-tree shortcut-ting algorithm from [Burkard et al. 1998]. It came as no surprise that this algorithmshowed higher approximation quality than all the tour constructing heuristics inReinelt’s experiment. Unfortunately, Reinelt’s experiment did not account for therunning time of the algorithms under investigation. The theoretical time com-plexity of the previous minimum-weight double-tree algorithm from [Burkard et al.1998] is O ( n + 2 d n ); in practice, our implementation of this algorithm exhibitedquadratic growth in running time on most instances. Both the theoretical and thepractical running times were relatively high, which raised some justifiable doubtsabout the overall superiority of the method.As it was expected, the introduction of the new efficient minimum-weight double-tree algorithm described in Section 2 significantly improved the running time in ourcomputational experiments. However, this improvement alone was not sufficientfor the algorithm to compete against the best existing tour-constructing heuristics.Therefore, we introduced two additional heuristic improvements, one aimed at in-creasing the algorithm’s speed, the other at improving its approximation quality.The first heuristic, aimed at speeding up the algorithm, is suggested by the well-known bounded neighbour lists [Johnson and McGeoch 2002, p. 408]. Given a tree,we define the tree distance between a pair of nodes a , b , as the number of edgeson the unique path from a to b in the tree. Given a parameter k , the depth- k list of node u includes all nodes in the subtree T ( u ) with the tree distance from u not exceeding k . The suggested heuristic improvement is to limit the searchacross a subtree rooted at u in (1)–(2) to a depth- k list of u for a suitably chosenvalue of k . Our experiments suggest that this approach improves the running timedramatically, without a significant negative effect on the approximation quality.The second heuristic, aimed at improving the algorithm’s approximation quality,works by expanding the space of the tours searched, in the hope of finding a better ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · solution in the larger space. Let T be a (not necessarily minimum) spanning tree,and let Λ( T ) be the set of all tours conforming to T , i.e. the exponential set of alltours considered by the double-tree algorithm. Our goal is to construct a new tree T , such that its node degrees are still bounded by a constant, but Λ( T ) ( Λ( T ).We refer to the new set of tours as an enlarged tour neighbourhood .Consider a node u in T , and suppose u has at least one child v which is not aleaf. We construct a new tree T from T by applying the degree-increasing operation ,which makes node v a leaf, and redefines all children of v to be children of u . It iseasy to check that any tour conforming to T also conforms to T . In particular, thenodes of T ( v ), which are consecutive in any conforming tour of T , are still allowedto be consecutive in any conforming tour of T . Therefore, Λ( T ) ⊆ Λ( T ). On theother hand, sequence w, u, v , where w is a child of v , is allowed by T but not by T . Therefore, Λ( T ) ( Λ( T ).Note that the degree-increasing operation cannot be performed partially: it wouldbe wrong to reassign only some, instead of all, children of node v to a new parent.To illustrate this statement, suppose that v has two children w and w , which areboth leaves. Let w be redefined as a new child of u . The sequence v, w , w isallowed by T but not by T , since it violates the requirement for v and w to beconsecutive. Therefore, Λ( T ) Λ( T ).We apply the degree-increasing heuristic as follows. Let D be a global parame-ter, not necessarily related to the maximum node degree in the original tree. Thedegree-increasing operation is performed only if the resulting new degree of vertex u would not exceed D . Given a tree, the degree increasing operation is applied re-peatedly to construct a new tree, obtaining an enlarged tour neighbourhood. In ourexperiments, we used breadth-first application of the degree increasing operationas follows:Root the minimum spanning tree at a node of degree 1;Let r ′ denote the unique child of the root;Insert all children of r ′ into queue Q ; while queue Q is not empty do extract node v from Q ;insert all children of v into Q ; if deg( parent ( v )) + deg( v ) ≤ D then redefine all children of v to be children of parent ( v )To incorporate the described heuristics, the minimum-weight double-tree algo-rithm from Section 2 was modified to take two parameters: the search depth k , andthe degree limit D . We refer to the double-tree algorithm with fixed parameters k and D as a double-tree heuristic DT D,k . We use DT without subscripts to denotethe original minimum-weight double-tree algorithm, equivalent to DT , ∞ .
4. COMPUTATIONAL EXPERIMENTS
We compared experimentally the efficiency of the original algorithm DT with theefficiency of double-tree heuristics DT
D,k for two different search depths k = 16 , D = 1 (no degree increasingoperation applied), 3, 4, 5. The case D = 2 is essentially equivalent to D = 1, andtherefore not considered. ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin
Size 1000 3162 10K 31K 100K 316K 1M 3MDT 7.36 7.82 8.01 8.19 8.39 8.40 8.41 –DT , , , , , , , , , , , , , , D,k on uniform Euclidean distances
The DIMACS Implementation Challenge [Johnson and McGeoch 2002] providedan excellent opportunity for testing and evaluating new approaches to the TSP.Website [DIMACS], created to support the Challenge, contains a wide range of testinstances and experimental data. In our computational experiments, we used uni-form random Euclidean instances with 1000 points (10 instances), 3162 points (fiveinstances), 10000 points (three instances), 31623 and 100000 points (two instancesof each size), 316228, 1000000, and 3168278 points (one instance of each size).For each heuristic, we consider both its approximation quality and running time.We say that one heuristic dominates another, if it is superior in both these respects.Following the approach of the DIMACS Challenge, approximation quality is mea-sured in terms of the approximate solution’s excess over the Held–Karp bound(the solution to the standard linear programming relaxation of the TSP), and therunning time in terms of the “normalised computation time” (see [Johnson and Mc-Geoch 2002], [DIMACS] for details). The experimental results, presented in Table I,clearly indicate that nearly all considered heuristics (excluding DT , ) dominateplain DT. Moreover, all these heuristics (again excluding DT , ) dominate DT oneach individual instance used in the experiment.For further comparison of the double-tree heuristics with existing tour-constructingheuristics, we chose DT , and DT , .The main part of our computational experiments consisted in comparing thedouble-tree heuristics against the most powerful existing tour-constructing heuris-tics. As a base for comparison, we chose the heuristics analysed in [Johnson and Heuristic DT , is omitted from Table I, since it does not give any noticeably better resultscompared to DT , . ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · Size 1000 3162 10K 31K 100K 316K 1M 3MRA + , , + , , A v e r ag ee x e ss o v e rt h e H e l d - K a r pb o und ( % ) b DT1;16 b Chr-G b A h b Sav b FI b Chr-S b RA+ b Chr-HK b DT5;16 b DT3;16
Fig. 4: Comparison between established heuristics and DT-heuristics on uniform Euclidean in-stances with 10000 points
ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin
McGeoch 2002], as well as two recent matching-based heuristics from [Kahng andReda 2004]. The experiments were performed on a Sun Systems Enterprise ServerE450, under SunOS 5.8, using the gcc 3.4.2 compiler.Table II shows the results of these experiments. Abbreviations in the table follow[Johnson and McGeoch 2002; Kahng and Reda 2004]:—RA + : Bentley’s random augmented addition heuristic;—Chr-S: the Christofides heuristic with standard shortcut, implemented by John-son and McGeoch (JM);—FI: Bentley’s farthest insertion heuristic;—Sav: saving heuristic, implemented by JM;—ACh: approximate Christofides heuristic, implemented by JM;—Chr-G: the Christofides heuristic with greedy shortcut, implemented by JM;—Chr-HK: the Christofides heuristic on Held–Karp trees instead of MST, imple-mented by Rohe;—MTS1, MTS3: “match twice and stitch” heuristics, implemented by Kahng andReda.As seen from the table, the average approximation quality of DT , turns out tobe higher than all classical heuristics considered in [Johnson and McGeoch 2002],except Chr-HK. Moreover, heuristic DT , dominates heuristics RA + , Chr-S,FI, Chr-G. Heuristic DT , dominates Chr-HK. Heuristic DT , also comparesvery favourably with MTS heuristics, providing similar approximation quality at asmall fraction of the running time. The above results show clearly that double-treeheuristics deserve a prominent place among the best tour-constructing heuristicsfor Euclidean TSP.The impressive success of double-tree heuristics must, however, be approachedwith some caution. Although the normalised time is an excellent tool for comparingresults reported in different computational experiments, it is only an approximateestimate of the exact running time. According to [Johnson and McGeoch 2002,page 377], “[this] estimate is still typically within a factor of two of the correcttime”. Therefore, as an alternative way of representing the results of computationalexperiments, we suggest a graph of the type shown in Figure 4, which comparesthe heuristics’ average approximation quality and running time on random uniforminstances with 10000 points. A normalised time t is represented by the interval[ t/ , t ]. The relative position of heuristics in the comparison and the dominancerelationships can be seen clearly from the graph. Results for other instance sizesand types are generally similar.Additional experimental results for clustered Euclidean instances are shown inTable III (with DT , replaced by DT , to illustrate more clearly the overalladvantage of DT-heuristics), and for TSPLIB instances in Table IV.While we have done our best to compare the existing and the proposed heuristicsfairly, we recognise that our experiments are not, strictly speaking, a “blind test”:we had the results of [Johnson and McGeoch 2002] in advance of implementingour method, and in particular of selecting the top DT-heuristics for comparison.However, we never consciously adapted our choices to the previous knowledge of ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · Size 1000 3162 10K 31K 100K 316KRA + , , + , , [Johnson and McGeoch 2002], and we believe that any subconscious effect of thisprevious knowledge on our experimental setup is negligible.
5. CONCLUSIONS AND OPEN PROBLEMS
In this paper, we have presented an improved algorithm for finding the minimum-weight double-tree shortcutting approximation for Metric TSP. We challenged our-selves to make the algorithm as efficient as possible. The improvement in timecomplexity from O ( n + 2 d n ) to O (4 d n ) (which implies O ( n ) for the EuclideanTSP) placed the minimum-weight double-tree shortcutting method as a peer in theset of the most powerful tour-constructing heuristics. It is known that most suchheuristics have theoretical time complexity O ( n ), and in practice often exhibitnear-linear running time. The minimum-weight double-tree method now also fitsthis pattern.While we have not been using the language of parameterised complexity [Downeyand Fellows 1998], we (and the previous work [Burkard et al. 1998]) have in factdemonstrated that the problem of finding the minimum-weight double-tree tour forMetric TSP is fixed-parameter tractable (where the maximum degree of the MSTis the relevant parameter). It would be interesting to see if this connection with ACM Journal Name, Vol. V, No. N, Month 20YY. · V. Deineko and A. Tiskin
Size 1000 3162 10K 31K 100KRA + , , + , , parameterised complexity theory can be extended further, e.g. by using any of theestablished techniques for designing fixed-parameter tractable algorithms.Our results should be regarded only as a first step in exploring new opportunities.Particularly, the minimum spanning tree is not the only possible choice of the initialtree. Instead, one can choose from a variety of trees, e.g. Held and Karp (1-)trees,approximations to Steiner trees, spanning trees of Delaunay graphs, etc. Thisvariety of choices merits a further detailed exploration.It is well-known that when the initial tree is a path, the resulting double-treetour neighborhood is the set of all pyramidal tours [Burkard et al. 1998]. In thiscase, a dozen of conditions on the distance matrix are known (see e.g. [Burkardet al. 1998]), which guarantee that the tour neighbourhood contains the absoluteminimum-weight tour. It may be possible to generalise this approach by identi-fying new special types of trees and conditions on the distance matrices, whichwould guarantee that the minimum-weight double-tree algorithm finds an absoluteminimum-weight tour. For more results on polynomial solvability of TSP with spe-cial conditions imposed on the distance matrix, see [Burkard et al. 1998; Deinekoet al. 2006]. ACM Journal Name, Vol. V, No. N, Month 20YY. ast minimum-weight double-tree shortcutting for Metric TSP · The minimum-weight shortcutting problem for the Christofides graph remainsNP-hard even in the planar Euclidean metric. However, our algorithm turns outto be applicable also to this problem on certain classes of instances. It can beshown that if the Christofides graph is a cactus (i.e. all its cycles are pairwise edge-disjoint), then the set of all its shortcuttings is a subset of the set of all double-treeshortcuttings. Therefore, our algorithm, as well as the algorithm of [Burkard et al.1998], can be used to find efficiently the minimum-weight shortcutting when theChristofides graph is a cactus. In particular, such a shortcutting can be found inpolynomial time in the planar Euclidean metric.Our efforts invested into theoretical improvements of the algorithm, supported bya couple of additional heuristic improvements, have borne the fruit: computationalexperiments with the minimum-weight double-tree algorithm show that it becomesone of the best known tour constructing heuristics. It appears that the double-treemethod is also well suited for local search improvements based of transformations oftrees and searching the corresponding tour neighborhoods. One can easily imaginevarious tree transformation techniques that could make our method even morepowerful.
6. ACKNOWLEDGEMENTS
The authors thank an anonymous referee of a previous version of this paper, whosedetailed comments helped to improve it significantly. The MST subroutine in ourcode is courtesy of the Concorde project [Concorde].
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