ON TREEWIDTH and STABLE MARRIAGE: PARAMETERIZED ALGORITHMS and HARDNESS RESULTS (COMPLETE CHARACTERIZATION)

Sushmita Gupta, Saket Saurabh, Meirav Zehavi

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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 (SESMI), Balanced Stable Marriage (BSMI), max-Stable Marriage with Ties (max-SMTI), and min-Stable Marriage with Ties (min-SMTI) problems. In this paper, we analyze these problems from the viewpoint of parameterized complexity. We conduct the first study of these problems in particular, and of problems related to Stable Marriage in general, with respect to the parameter treewidth. The motivation behind the choice of treewidth is threefold. First, several problems in social choice theory have already been studied with respect to treewidth. The networks relevant to these problems (say, social networks) are clearly also relevant to Stable Marriage. Thus, the motivation underlying these studies directly extends to our study. Second, empirical studies of the treewidth of several types of networks relevant to Stable Marriage have also already been undertaken, identifying that some of these networks indeed have a treelike structure. Third, treewidth is the most well studied structural parameter in parameterized complexity. We design optimal parameterized algorithms for all four problems under the treewidth of both their primal graphs and rotation digraphs. 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 unless the exponential-time hypothesis is false, all of these algorithms are optimal. Next, we study the treewidth tw of the rotation digraph. In this context, max-SMTI and min-SMTI are not defined. For both SESMI and BSMI, we design (highly nontrivial) algorithms that run in time 2twnO(1). Then, for both SESMI and BSMI, we prove that unless the strong exponential-time hypothesis 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 Stable Marriage with respect to treewidth.

Original languageEnglish
Pages (from-to)596-681
Number of pages86
JournalSIAM Journal on Discrete Mathematics
Volume36
Issue number1
DOIs
StatePublished - 1 Jan 2022

Keywords

  • parameterized complexity
  • stable marriage
  • treewidth

ASJC Scopus subject areas

  • General Mathematics

Fingerprint

Dive into the research topics of 'ON TREEWIDTH and STABLE MARRIAGE: PARAMETERIZED ALGORITHMS and HARDNESS RESULTS (COMPLETE CHARACTERIZATION)'. Together they form a unique fingerprint.

Cite this