TY - JOUR

T1 - On Treewidth and Stable Marriage.

AU - Gupta, Sushmita

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2017

Y1 - 2017

N2 - Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth tw of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time nO(tw), we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth tw of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time 2twnO(1). Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time (2−ϵ)twnO(1) do not exist for any fixed ϵ>0. We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth.

AB - Stable Marriage is a fundamental problem to both computer science and economics. Four well-known NP-hard optimization versions of this problem are the Sex-Equal Stable Marriage (SESM), Balanced Stable Marriage (BSM), max-Stable Marriage with Ties (max-SMT) and min-Stable Marriage with Ties (min-SMT) problems. In this paper, we analyze these problems from the viewpoint of Parameterized Complexity. We conduct the first study of these problems with respect to the parameter treewidth. First, we study the treewidth tw of the primal graph. We establish that all four problems are W[1]-hard. In particular, while it is easy to show that all four problems admit algorithms that run in time nO(tw), we prove that all of these algorithms are likely to be essentially optimal. Next, we study the treewidth tw of the rotation digraph. In this context, the max-SMT and min-SMT are not defined. For both SESM and BSM, we design (non-trivial) algorithms that run in time 2twnO(1). Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time (2−ϵ)twnO(1) do not exist for any fixed ϵ>0. We thus present a comprehensive, complete picture of the behavior of central optimization versions of Stable Marriage with respect to treewidth.

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JO - arxiv cs.DS

JF - arxiv cs.DS

IS - abs/1707.05404

ER -