2020/07/21 by Kristóf Bérczi, Bérczi, Kristóf, Matthias Mnich +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Approximation algorithm #Bounded function #Combinatorics #Complexity and Algorithms in Graphs #Computer science #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #Discrete mathematics #FOS: Computer and information sciences #Mathematical optimization #Mathematics #Matroid #Metric (unit) #Optimization and Search Problems #Path (computing) #cs.DM #cs.DS
paper · pdf · doi:10.48550/arxiv.2007.11389
published in arXiv (Cornell University) (Cornell University) · arXiv admin note: text overlap with arXiv:1911.09890
arxiv created 2020/07/21 · openalex publication_date 2020/07/21 · arxiv updated 2020/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 Király, 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.