A 3/2-Approximation for the Metric Many-Visits Path TSP

Abstract

In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or costs) $c(uv)$, and moreover each city $v$ comes with an associated positive integer request $r(v)$. The goal is to find a minimum-cost path, starting at city $s$ and ending at city $t$, that visits each city $v$ exactly $r(v)$ times. We present a $3/2$-approximation algorithm for the metric Many-visits Path TSP that runs in time polynomial in $n$ and polylogarithmic in the requests $r(v)$. Our algorithm can be seen as a generalization of the $3/2$-approximation algorithm for Path TSP by Zenklusen [Proceedings of SODA, 2019, pp. 1539--1549], which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP. One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree-bounded multigraph of minimum cost in an undirected graph with edge costs. We tackle this problem by generalizing a fundamental result of Király, Lau, and Singh [Combinatorica, 32 (2012), pp. 705--720] on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for generalized polymatroids, even allowing element multiplicities. Our result directly yields a $3/2$-approximation to the metric Many-visits TSP, as well as a $3/2$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.