On Treewidth and Stable Marriage [article]

Sushmita Gupta, Saket Saurabh, Meirav Zehavi
2017 arXiv   pre-print
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
more » ... y 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 n^O(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 2^twn^O(1). Then, for both SESM and BSM, we also prove that unless SETH is false, algorithms that run in time (2-ϵ)^twn^O(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.
arXiv:1707.05404v1 fatcat:wvzfalp4rzd4nn2atrxfzj47j4