Operations Research Letters an Improved Upper Bound for the TSP in Cubic 3-Edge-Connected Graphs

Operations Research Letters Operations Research Letters 33 (2005) 467–474 www.elsevier.com/locate/orl An improvedupper boundfor the TSP in cubic 3-edge-connected graphs DavidGamarnik b, Moshe Lewensteina,∗,1, Maxim Sviridenkob aDepartment of Computer Science, Bar Ilan University, Room 323, Mailbox 50, Ramat Gan 52900, Israel bIBM T. J. Watson Research Center, Yorktown Heights, P.O. Box 218, NY 10598, USA Received20 January 2004; accepted3 September 2004 Abstract We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions andthe subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general. © 2004 Elsevier B.V. All rights reserved. Keywords: Approximation algorithms; Travelling salesman problem; Graphs; Regular graphs 1. Introduction In general the TSP problem is NP-complete andnot approximable to any constant. Even the metric variant In this paper we consider the well-known travelling of TSP, where the edge weights of Gc form a metric salesman problem (TSP). Given a complete undirected over the vertex set, is known to be MAXSNP-hard [8]. graph Gc = (V, Ec) with nonnegative edge weights, However, Christofides [5] introduced an elegant ap- the goal is to finda shortest (accordingto the weights) proximation algorithm for metric TSP that produces closedtour visiting each vertex exactly once. A closed a solution with an approximation factor 3/2. Almost tour visiting each vertex exactly once is also known three decades later, there is still no approximation al- as a Hamiltonian cycle. gorithm with a better factor than Christofides’ 3/2 factor. A detailedsurvey of the results andthe history of TSP appears in [7,6]. ∗ Corresponding author. Tel.: +972 3 531 7668; fax: +972 3 736 Let G be an arbitrary graph with edge weights that 0498. do not violate the triangle inequality. A metric com- E-mail addresses: [email protected] (D. Gamarnik), pletion on G is the completion of G into a complete [email protected] (M. Lewenstein), [email protected] graph Gc such that each added edge (u, v) is weighted (M. Sviridenko). G 1 The research in this paper was done when the author was a in a manner so that c forms a metric. A shortest path postdoc at IBM T.J. Watson Research Center. metric completion is a metric completion for which 0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.09.005 468 D. Gamarnik et al. / Operations Research Letters 33 (2005) 467–474 each edge added is assigned the weight of the shortest This leads to the following conjecture with regard path between u and v. Since we only consider short- to . est path metric completions in this paper, we will re- fer to them as metric completions. In general, in this Conjecture. For metric TSP, the integrality gap for paper we study the case when G is a cubic 3-edge SER is 4/3. connectedgraph and Gc is its metric completion. All edge weights of the original graph are 1. There have been numerous attempts to prove this One of the classical integer programming formula- conjecture true; however, so far these attempts have tions of the TSP problem is as follows: been unsuccessful. In fact, not only has the conjecture not been proven so far, there has not even been min w(u,v)x(u,v) (1) an improvement over the 3/2 factor. This has raised (u,v)∈E / c the counterquestion: perhaps 3 2 is the best factor achievable for the metric TSP problem. Or a some- x(u,v) = 2 for all v ∈ V, (2) what milder variant of the question is, perhaps for (u,v)∈Ec = IP∗ /LP ∗ , = 3/2 is the best achievable. x ∅ = S ⊂ V, (u,v) 2 for all (3) An optimal solution to the subtour elimination u∈S,v∈V \S LP is closely relatedto k-regular k-edge-connected x(u,v) ∈{0, 1} for all u, v ∈ V. (4) multigraphs in the following sense. Given an optimal solution to the LP, we define the number D to be the We denote optimal solutions of the above integer pro- ∗ ∗ smallest common multiplier of the variables xe for gram with IP andits value with IP∗ . As men- ∗ all edges e, i.e. D satisfies that for each edge e, Dxe tionedabove, finding IP∗ is NP-complete. Changing is integer. On the vertex set V we define a multigraph requirement (4) to require that x(u,v) 0 yields a linear ∗ with Dxe edges between vertices u and v where programming relaxation of the integer program also e = (u, v). This multigraph is 2D-regular and2 D- known as the subtour elimination relaxation (SER for edge-connected by the constraints of the LP. Consider short). the original graph G, which was later completedto a Despite the drawback that there are an exponential metric as a TSP instance. Say the weights of all edges number of constraints (of type (3)), SER can neverthe- were 1, i.e. it was unweighted. If the optimal value of less be solvedin polynomial time since each iteration the LP was n, i.e. G was “fractionally” Hamiltonian, of the separation oracle is one mincut computation. We ∗ then it wouldbe enough to finda tour of value n denote with LP the solution to SER andwith ∗ LP with < 3/2 to improve on Cristofides algorithm for its value. A measure for the quality of approximation that special (yet very general) case. of an SER solution is the worst-case ratio (integrality We consider probably the simplest class of graphs gap) satisfying the above properties, the cubic (that is, AP X 3-regular) 3-edge-connected simple graphs with = max , LP ∗ weights 1. The metric completions of these graphs have weights that can be substantially large. It is where AP X is a value of an SER solution andthe easily seen that LP ∗ = n on this class of graphs. maximum is over all instances of metric TSP. Indeed, consider the following solution: x(u,v) = 2/3 The best-known upper boundto date [10,11] shows if (u, v) ∈ E and x(u,v) = 0 otherwise. The feasibility that 3/2. However, no example showing to be of this solution follows from the fact that graph G is 3/2 is known. The well-known, andcurrently best, cubic and 3-edge-connected. Since the value of this lower boundon appears in Fig. 1. The full example solution is n and n is a trivial lower boundon LP ∗ , is a metric completion on this graph. To see the 4/3 this solution is optimal. gap, set x(u,v) to be 1/2 for the weight 2 edges in the Thus, to improve the boundon the value of the op- graph and x(u,v) to be 1 for the weight 1 edges. For all timal Hamiltonian cycle, it is enough to prove that other edges, set x(u,v) = 0. In Fig. 1 the optimal tour there is a Hamiltonian cycle of weight at most n in is shown in the right-handfigure. Gc for < 3/2. In this paper, we design a polynomial D. Gamarnik et al. / Operations Research Letters 33 (2005) 467–474 469 111 ... 111 ... 2 2 2 11 11 2 2 2 ... ... 111 111 Fig. 1. An example for which is 4/3. time algorithm which finds a Hamiltonian cycle in Gc Also say we have a spanning tree T = (V, E) on of weight at most (3/2 − 5/389)n. Approximation al- a connectedgraph G. We call the multigraph (V, E ), gorithms with performance guarantees better than 3/2 where E contains two copies of each edge of E,a (or even PTASes) for other special classes of metric double spanning tree. We say that we double T when spaces can be foundin [1–3,8]. we take a double spanning tree according to T. There are several directions for possible improve- A cycle cover is a collection of node disjoint cycles ments of our results. It is conceivable that the im- covering all nodes in a graph. The following result provements over 3/2 approximations can be obtained from 1891 is due to Petersen [9]. Its proof can be for general r-regular graphs. Another possibility is to foundin Behzad,ChartrandandLesniak-Foster [4]. obtain improvedresults for sparse graphs, say graphs with maximum degree 3. At this point, we cannot even Lemma 1 (Petersen [9]). Let G=(V, E) be a 2-edge- lift the 3-connectivity restriction. connected cubic graph. The edge set E can be partitioned into a perfect matching M and a cycle cover C. 2. Preliminaries Note that both problems of finding a perfect matching andfindinga cycle cover are polynomially solv- Consider a cubic 3-edge-connected graph G and able. Using this result we show the following lemma. the corresponding complete weighted graph Gc.Any G=(V, E) Hamiltonian cycle in Gc corresponds to some closed Lemma 2. Let be a 3-edge-connected cu- path in G of the same weight that visits all vertices bic graph. The edge set E can be partitioned into a at least once and, conversely, any such path corre- perfect matching M and a triangle-free cycle cover C sponds to a Hamiltonian cycle in Gc of the same (that is a cycle cover which does not contain cycles of weight.

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