On Treewidth and Stable Marriage: Parameterized Algorithms and Hardness Results (Complete Characterization)

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 ${\mathtt{tw}}$ 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 $n^{{\mathcal{O}}({\mathtt{tw}})}$, we prove that unless the exponential-time hypothesis is false, all of these algorithms are optimal. Next, we study the treewidth ${\mathtt{tw}}$ 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 $2^{{\mathtt{tw}}}n^{{\mathcal{O}}(1)}$. Then, for both SESMI and BSMI, we prove that unless the strong exponential-time hypothesis is false, algorithms that run in time $(2-\epsilon)^{{\mathtt{tw}}}n^{{\mathcal{O}}(1)}$ do not exist for any fixed $\epsilon>0$. We thus present a comprehensive, complete picture of the behavior of Stable Marriage with respect to treewidth.