Better s-t-Tours by Gao Trees

Abstract

We consider the s-t-path TSP: given a finite metric space with two elements s and t, we look for a path from s to t that contains all the elements and has minimum total distance. We improve the approximation ratio for this problem from 1.599 to 1.566. Like previous algorithms, we solve the natural LP relaxation and represent an optimum solution x * as a convex combination of spanning trees. Gao showed that there exists a spanning tree in the support of x * that has only one edge in each narrow cut (i.e., each cut C with x * (C) < 2).Our main theorem says that the spanning trees in the convex combination can be chosen such that many of them are such "Gao trees".

A 1.5-Approximation for Path TSP

“…Vygen [25] showed that further improvement is possible by first reassembling the trees appearing in a convex decomposition of x * before sampling, to obtain desirable properties for cheaper parity correction. An additional strengthening was achieved by Gottschalk and Vygen [8], who derived a beautiful structural result showing that a very well-structured convex decomposition into trees is possible by generalizing a result of Gao [6]. 5 Moreover, Sebő and van Zuylen [20] modified the general connectivity plus parity correction approach by first deleting some edges from the initial spanning tree, thus of characteristic vectors χ J i ∈ {0, 1} E of Q-joins Ji ⊆ E for i ∈ {1, .…”

“…2 Except for the last result listed in the table, all other results also imply an equivalent upper bound on the integrality gap of the Held-Karp relaxation, which is known to be at least 1.5 (see, e.g., [2]). Reference Ratio Hoogeveen [9] 1.667 An, Kleinberg, and Shmoys [2] 1.618 Sebő [19] 1.6 Vygen [25] 1.599 Gottschalk and Vygen [8] 1.566 Sebő and van Zuylen [20] 1.529 Traub and Vygen [22] 1.5 + ǫ Table 1: Previous approximation guarantees (rounded) for Path TSP. 2 On a high level, all of these results build upon the same original approach of Christofides for TSP, which is based on first obtaining a connected graph-through a spanning tree-and then performing parity correction of the degrees by adding additional edges.…”

An improved upper bound on the integrality ratio for thes–t-path TSP

“…, H k . All previous analyses, like (3), computed an upper bound on k j=1 p j c(H j ). Instead, we will compute a weighted average with different weights, giving a higher weight to tours resulting from early trees.…”

“…For this LP (see (1) below) the conjectured integrality ratio is 3 2 , which is asymptotically attained by simple examples. Better and better upper bounds have been shown [1,6,9,3,7]. The previously best-known upper bound by Sebő and van Zuylen [7] is 3 2 + 1 34 > 1.5294.…”

“…Better and better upper bounds have been shown [1,6,9,3,7]. The previously best-known upper bound by Sebő and van Zuylen [7] is 3 2 + 1 34 > 1.5294. We improve the analysis of their algorithm and show that the integrality ratio is smaller than 1.5284.…”