An LP-based -approximation algorithm for the path graph Traveling Salesman Problem
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 . Hoogeveen [8] extended the algorithm to the metric path TSP, and proved an approximation guarantee of . This had been the best approximation factor for decades until a recent paper [1, An, Kleinberg, and Shmoys] improved on the approximation guarantee and presented an algorithm that achieves an approximation guarantee of . Most recently, [11, Sebő] further improved the approximation factor to 1.6.
For the path graph TSP, a special case of the metric path TSP, [2, An and Shmoys] provided a sightly improved performance guarantee of . The paper [9, Mömke and Svensson] gave a 1.586-approximation algorithm for the path graph TSP. [10, Mucha] improved the analysis of [9] and obtained a approximation guarantee for any for the path graph TSP. Recently, [12, Sebő and Vygen] gave the first 1.5-approximation algorithm for the path graph TSP by using ear decomposition and matroid intersection.
In this paper, we present a new 1.5-approximation algorithm for the 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 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 be a connected graph with unit cost on each edge. Let be two given vertices in . Consider the metric completion of , where is the cost function on each edge in such that is the minimal cost of any path in . The path graph TSP is to find a minimum cost Hamiltonian path from to in with edge costs . Denote the cost of this path by .
For any vertex subset , we define . If there is no ambiguity, we use
LP-based -approximation algorithm
In this section, we give an LP-based -approximation algorithm for the 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 .
Proof Compute the Gomory–Hu tree for the terminal vertex set with respect to the capacity (see [5, Section 3.5.2]). After that, for each edge of the 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.
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