Finding Robust Solutions to Stable Marriage

Abstract

We study the notion of robustness in stable matching problems. We first define robustness by introducing (a, b)-supermatches. An (a, b)-supermatch is a stable matching in which if a pairs break up it is possible to find another stable matching by changing the partners of those a pairs and at most b other pairs. In this context, we define the most robust stable matching as a (1, b)-supermatch where b is minimum. We show that checking whether a given stable matching is a (1, b)-supermatch can be done in polynomial time. Next, we use this procedure to design a constraint programming model, a local search approach, and a genetic algorithm to find the most robust stable matching. Our empirical evaluation on large instances show that local search outperforms the other approaches.

Complexity Study for the Robust Stable Marriage Problem

“…Formally, a stable matching M is said to be (a, b)-supermatch if for any set Ψ ⊆ M of a stable pairs that are not fixed, there exists a stable matching [7]. It is important to note here that the Rural Hospitals Theorem (see Theorem 1.4.2 in [12]) states that the same set of agents are assigned in all stable matchings.…”

“…Let us illustrate these terms on a sample SM instance specified by the preference lists of 7 men/women in Table 1 given by Genc et. al [7]. For the sake of clarity, each man m i is denoted with i and each woman w j with j in the preference lists.…”

“…In this paper, we focus on the robustness notion proposed by Genc et al [9,7] related to the concept of an (a, b)-supermatch. Informally, an (a, b)supermatch is a stable matching such that if any (non-fixed) a men break-up, it is possible to find another stable matching by changing the partners of those a men and at most b others.…”

On the Complexity of Robust Stable Marriage

“…In the context of Stable Marriage, the purpose is to find a matching M between men and women such that no pair man, woman prefer each other to their situations in M . The authors of [7] introduced the notion of (a, b)-supermatch as a measure of robustness. An (a, b)-supermatch is a stable matching such that if any a agents (men or woman) break up it is possible to find another stable matching by changing the partners of those a agents with also changing the partners of at most b others.…”

“…Both finding an (a, b)-supermodel and an (a, b)-super solution are shown to be N P-complete. However, they leave the complexity of this problem open [7].…”