Balanced Stable Marriage: How Close is Close Enough?

TL;DR

This paper explores the computational complexity of the Balanced Stable Marriage problem, focusing on parameterized complexity and identifying conditions under which the problem is tractable or intractable.

Contribution

It introduces natural parameterizations based on satisfaction guarantees, providing a clear boundary between feasible and infeasible instances.

Findings

Parameterizations align with scenarios where stable matchings are acceptable to both parties.

The study delineates the boundary between tractability and intractability based on satisfaction levels.

The problem's complexity varies significantly depending on the satisfaction target values.

Abstract

The Balanced Stable Marriage problem is a central optimization version of the classic Stable Marriage problem. Here, the output cannot be an arbitrary stable matching, but one that balances between the dissatisfaction of the two parties, men and women. We study Balanced Stable Marriage from the viewpoint of Parameterized Complexity. Our "above guarantee parameterizations" are arguably the most natural parameterizations of the problem at hand. Indeed, our parameterizations precisely fit the scenario where there exists a stable marriage that both parties would accept, that is, where the satisfaction of each party is "close" to the best it can hope for. Furthermore, our parameterizations accurately draw the line between tractability and intractability with respect to the target value.