Degree-Bounded Polymatroids, with Applications to the Many-Visits TSP

Speaker:
Matthias Mnich - COLLOQUIUM LECTURE
Date:
Tuesday, 3.12.2019, 14:30
Place:
Room 337 Taub Bld.
Affiliation:
University of Bonn
Host:
Hadas Shachnai

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 f(e) and g(e) for each hyperedge e. The objective is to find a minimum-cost basis B such that f(e) <= |B \cap e| <= g(e) for each hyperedge e. Kiraly, Lau and Singh (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 2\Delta-1, where \Delta is the maximum degree of the hypergraph. We consider an extension of the matroid basis problem to generalized polymatroids, and additionally allow element multiplicities. Building on the approach of Kiraly, Lau and Singh, we provide an algorithm for finding a solution of cost at most the optimum value having the same additive approximation guarantee. As an application, we develop a 1.5-approximation for the metric many-visits TSP, where the goal is to find a minimum-cost tour that visits each city v a positive number r_v of times. Our approach combines our algorithm for the Lower Bounded Degree Generalized Polymatroid Basis Problem with Multiplicities with the principle of Christofides' algorithm from 1976 for the (single-visit) metric TSP, whose approximation guarantee it matches. We also present a new algorithm for the general many-visits TSP without any metric assumptions on the edge cost. For this problem, Cosmadakis and Papadimitriou (SICOMP, 1984) provided an algorithm that finds an optimal tour in time and space that depends superexponentially on the number of cities. We give an improved algorithm which simultaneously improves the time complexity to single-exponential, and the space complexity to polynomial. Assuming the Exponential Time Hypothesis, the run time of our algorithm is asymptotically optimal. Our algorithm is arguably simpler than the one by Cosmadakis and Papadimitriou. (Based on joint work with Kristof Berczi, Andre Berger, Tamas Kiraly, Laszlo Kozma, Gyula Pap and Roland Vincze) Short Bio: ========== Matthias Mnich is an assistant professor of quantitative economics at Maastricht University, The Netherlands, currently on leave for research in theoretical computer science at the Rheinische Friedrich-Wilhelms Universitaet Bonn, Germany. He obtained his PhD in 2010 at Eindhoven University of Technology, for which he received the Philips Prize 2010. Since then he held positions at UC Berkeley, USA and the Max Planck Institute for Computer Science, and a visiting professorship at TU Darmstadt, Germany. His research interests are in combinatorial algorithms, social choice theory, algorithms for big data, and algorithms in artificial intelligence. ================================================== Refreshments will be served from 14:15 Lecture starts at 14:30

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