Elsevier

Operations Research Letters

Volume 41, Issue 6, November 2013, Pages 615-617
Operations Research Letters

An LP-based 32-approximation algorithm for the st path graph Traveling Salesman Problem

https://doi.org/10.1016/j.orl.2013.08.006Get rights and content

Abstract

We design a new LP-based algorithm for the st path graph traveling salesman problem, which achieves in a simpler way the best-known approximation factor of 1.5, a result of Sebő and Vygen. The algorithm is based on the idea of narrow cuts due to An, Kleinberg, and Shmoys. It answers a call for a simple 32-approximation algorithm from Sebő.

Introduction

The metric traveling salesman problem (TSP) is one of the best-known problems in the area of combinatorial optimization. For the metric TSP, Christofides  [4] presented an algorithm that achieves an approximation guarantee of 32. Hoogeveen  [8] extended the algorithm to the metric st path TSP, and proved an approximation guarantee of 53. This had been the best approximation factor for decades until a recent paper  [1, An, Kleinberg, and Shmoys] improved on the 53 approximation guarantee and presented an algorithm that achieves an approximation guarantee of 1+521.61803. Most recently,  [11, Sebő] further improved the approximation factor to 1.6.

For the st path graph TSP, a special case of the metric st path TSP,  [2, An and Shmoys] provided a sightly improved performance guarantee of (53ϵ). The paper  [9, Mömke and Svensson] gave a 1.586-approximation algorithm for the st path graph TSP.  [10, Mucha] improved the analysis of  [9] and obtained a 1912+ϵ1.58333+ϵ approximation guarantee for any ϵ>0 for the st path graph TSP. Recently,  [12, Sebő and Vygen] gave the first 1.5-approximation algorithm for the st path graph TSP by using ear decomposition and matroid intersection.

In this paper, we present a new 1.5-approximation algorithm for the st path graph TSP. Compared with the algorithm from [12, Sebő and Vygen], our algorithm and its analysis are much simpler. In  [11], Sebő posed an open question on applying the “Best of Many Christofides” algorithm in  [1] to achieve the best approximation guarantees known for the graphic special cases of TSP and its variants. Although our paper does not address this specific question, it answers a call for a simple 1.5-approximation algorithm by computing narrow cuts; the notion of narrow cuts for st path TSP was introduced by  [1, An, Kleinberg, and Shmoys] for their analysis. The key point of our algorithm is to find a minimum spanning tree that intersects every narrow cut in an odd number of edges. Such a tree guarantees that the number of edges fixing the wrong degree vertices is at most half of the optimal value of the linear programming relaxation. Finally, the union of the spanning tree and the added edges provide us the 1.5-approximation guarantee. The detailed description of our algorithm is presented in Section  3. The graphic property is used to guarantee that the cost of the spanning tree we find is bounded by the optimum of the linear programming relaxation. However, for the general metric case, there exists an example such that the cost of the spanning tree in our algorithm is strictly larger than the optimum of the linear programming relaxation.

Section snippets

Preliminaries

Let G=(V,E) be a connected graph with unit cost on each edge. Let s,t be two given vertices in G. Consider the metric completion (G,c) of G, where c is the cost function on each edge e=(u,v) in G such that ce is the minimal cost of any uv path in G. The st path graph TSP is to find a minimum cost Hamiltonian path from s to t in G with edge costs c. Denote the cost of this path by OPT(G).

For any vertex subset ϕSV, we define δG(S)={(u,v)E:uS,vS}. If there is no ambiguity, we use δ(S)

LP-based 32-approximation algorithm

In this section, we give an LP-based 32-approximation algorithm for the st path TSP. Before stating the algorithm, we need some lemmas. The first lemma is well known, but we include the proof for the sake of completeness.

Lemma 3.1

There is a polynomial-time combinatorial algorithm to find all narrow cuts S1,S2,,Sk.

Proof

Compute the Gomory–Hu tree for the terminal vertex set V with respect to the capacity x (see  [5, Section 3.5.2]). After that, for each edge of the st path in the Gomory–Hu tree, check the

Acknowledgments

I am grateful to Joseph Cheriyan for stimulating discussions and indispensable help, and to Zachary Friggstad, Laura Sanità, and Chaitanya Swamy for their useful comments. I thank the anonymous reviewers for their constructive suggestions, which helped to improve the quality of this paper.

References (12)

  • J. Hoogeveen

    Analysis of Christofides’ heuristic: some paths are more difficult than cycles

    Oper. Res. Lett.

    (1991)
  • H.-C. An, R. Kleinberg, D.B. Shmoys, Improving Christofides’ algorithm for the s–t path TSP, in: STOC, 2012, pp....
  • H.-C. An, D.B. Shmoys, LP-based approximation algorithms for traveling salesman path problems. CoRR...
  • J. Cheriyan, Z. Friggstad, Z. Gao, Approximating minimum-cost connected T-joins, in: APPROX-RANDOM, 2012, pp....
  • N. Christofides

    Worst-case analysis of a new heuristic for the travelling salesman problem, Tech. Rep. 388

    (1976)
  • W.J. Cook et al.

    Combinatorial Optimization

    (1998)
There are more references available in the full text version of this article.

Cited by (20)

  • REDUCING PATH TSP TO TSP

    2022, SIAM Journal on Computing
View all citing articles on Scopus
View full text