On the Complexity of Robust Stable Marriage

TL;DR

This paper investigates the computational complexity of finding robust stable matchings that can be efficiently repaired after disruptions, proving NP-Completeness via SAT formulation and equivalence to specific instances.

Contribution

It introduces a SAT-based NP-Completeness proof for the (a,b)-supermatch problem in robust stable marriage and establishes equivalence for the (1,1)-supermatch case.

Findings

Deciding existence of an (a,b)-supermatch is NP-Complete.

A SAT formulation for the problem is NP-Complete.

Equivalence between SAT formulation and (1,1)-supermatch on certain instances.

Abstract

Robust Stable Marriage (RSM) is a variant of the classical Stable Marriage problem, where the robustness of a given stable matching is measured by the number of modifications required for repairing it in case an unforeseen event occurs. We focus on the complexity of finding an (a,b)-supermatch. An (a,b)-supermatch is defined as a stable matching in which if any 'a' (non-fixed) men/women break up it is possible to find another stable matching by changing the partners of those 'a' men/women and also the partners of at most 'b' other couples. In order to show deciding if there exists an (a,b)-supermatch is NP-Complete, we first introduce a SAT formulation that is NP-Complete by using Schaefer's Dichotomy Theorem. Then, we show the equivalence between the SAT formulation and finding a (1,1)-supermatch on a specific family of instances.