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

TL;DR

This paper introduces a polynomial-time 3/2-approximation algorithm for the metric Many-visits Path TSP, generalizing previous results and addressing a long-standing open problem in route optimization.

Contribution

It presents the first efficient 3/2-approximation algorithm for the metric Many-visits Path TSP, extending techniques from Path TSP and matroid theory.

Findings

Achieves a 3/2-approximation ratio for the problem.

Develops a polynomial-time algorithm for minimum cost connected, degree-bounded multigraphs.

Generalizes a key matroid basis result to polymatroids with element multiplicities.

Abstract

In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or cost) $c(uv)$, 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 $\frac32$-approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in $n$ and poly-logarithmic in the requests $r(v)$. Our algorithm can be seen as a far-reaching generalization of the $\frac32$-approximation algorithm for Path TSP by Zenklusen (SODA 2019), 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. We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for general polymatroids, even allowing element multiplicities. Our result directly yields a $\frac32$-approximation to the metric Many-visits TSP, as well as a $\frac32$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.