Title: On Treewidth and Stable Marriage,
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 (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 $\mathtt{tw}$
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 $n^{O(\mathtt{tw})}$, we prove that all of these algorithms
are likely to be essentially optimal. Next, we study the treewidth
$\mathtt{tw}$ 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 $2^{\mathtt{tw}}n^{O(1)}$. Then, for both SESM and BSM, we also
prove that unless SETH is false, algorithms that run in time
$(2-\epsilon)^{\mathtt{tw}}n^{O(1)}$ do not exist for any fixed $\epsilon>0$.
We thus present a comprehensive, complete picture of the behavior of central
optimization versions of Stable Marriage with respect to treewidth.