Degree-Bounded Generalized Polymatroids and Approximating the Metric Many-Visits TSP
Krist\'of B\'erczi, Andr\'e Berger, Matthias Mnich, Roland, Vincze

TL;DR
This paper extends matroid basis problems to generalized polymatroids with element multiplicities, providing approximation algorithms that improve solutions for complex combinatorial optimization problems like the metric Many-Visits TSP.
Contribution
It introduces a new algorithm for bounded degree g-polymatroid problems with multiplicities, achieving the same approximation guarantees as prior matroid-based methods.
Findings
Provides a 1.5-approximation for the metric Many-Visits TSP.
Extends previous algorithms to g-polymatroids with multiplicities.
Maintains approximation guarantees similar to Christofides' algorithm.
Abstract
In the Bounded Degree Matroid Basis Problem, we are given a matroid and a hypergraph on the same ground set, together with costs for the elements of that set as well as lower and upper bounds and for each hyperedge . The objective is to find a minimum-cost basis such that for each hyperedge . Kir\'aly et al. (Combinatorica, 2012) provided an algorithm that finds a basis of cost at most the optimum value which violates the lower and upper bounds by at most , where is the maximum degree of the hypergraph. When only lower or only upper bounds are present for each hyperedge, this additive error is decreased to . We consider an extension of the matroid basis problem to generalized polymatroids, or g-polymatroids, and additionally…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
