240 research outputs found

    Improving on Best-of-Many-Christofides for TT-tours

    Full text link
    The TT-tour problem is a natural generalization of TSP and Path TSP. Given a graph G=(V,E)G=(V,E), edge cost c:E→R≥0c: E \to \mathbb{R}_{\ge 0}, and an even cardinality set T⊆VT\subseteq V, we want to compute a minimum-cost TT-join connecting all vertices of GG (and possibly containing parallel edges). In this paper we give an 117\frac{11}{7}-approximation for the TT-tour problem and show that the integrality ratio of the standard LP relaxation is at most 117\frac{11}{7}. Despite much progress for the special case Path TSP, for general TT-tours this is the first improvement on Seb\H{o}'s analysis of the Best-of-Many-Christofides algorithm (Seb\H{o} [2013])

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

    Get PDF
    In the Many-visits Path TSP, we are given a set of nn cities along with their pairwise distances (or cost) c(uv)c(uv), and moreover each city vv comes with an associated positive integer request r(v)r(v). The goal is to find a minimum-cost path, starting at city ss and ending at city tt, that visits each city vv exactly r(v)r(v) times. We present a 32\frac32-approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in nn and poly-logarithmic in the requests r(v)r(v). Our algorithm can be seen as a far-reaching generalization of the 32\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 32\frac32-approximation to the metric Many-visits TSP, as well as a 32\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.Comment: arXiv admin note: text overlap with arXiv:1911.0989
    • …
    corecore