240 research outputs found
Improving on Best-of-Many-Christofides for -tours
The -tour problem is a natural generalization of TSP and Path TSP. Given a
graph , edge cost , and an even
cardinality set , we want to compute a minimum-cost -join
connecting all vertices of (and possibly containing parallel edges).
In this paper we give an -approximation for the -tour
problem and show that the integrality ratio of the standard LP relaxation is at
most . Despite much progress for the special case Path TSP, for
general -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
In the Many-visits Path TSP, we are given a set of cities along with
their pairwise distances (or cost) , and moreover each city comes
with an associated positive integer request .
The goal is to find a minimum-cost path, starting at city and ending at
city , that visits each city exactly times.
We present a -approximation algorithm for the metric Many-visits
Path TSP, that runs in time polynomial in and poly-logarithmic in the
requests .
Our algorithm can be seen as a far-reaching generalization of the
-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 -approximation to the metric
Many-visits TSP, as well as a -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
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