Preference Swaps for the Stable Matching Problem

Abstract

An instance I of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in I is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of I. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum number of swaps required to turn a given maximal matching into a stable matching. To generalize this result to the many-to-many version of SMP, we introduce a new representation of SMP as an extended bipartite graph and reduce the problem to submodular minimization. It is a natural problem to establish computational complexity of deciding whether at most k swaps are enough to turn I into an instance where one of the maximum matchings is stable. Using a hardness result of Gupta et al. (2020), we prove that it is NP-hard to decide whether at most k swaps are enough to turn I into an instance with a stable perfect matching. Moreover, this problem parameterized by k is W[1]-hard. We also obtain a lower bound on the running time for solving the problem using the Exponential Time Hypothesis.

Manipulating the outcome of stable matching and roommates problems

“…They examined several variants in terms of manipulative actions and manipulation goals in bipartite preference systems. Later, Eiben et al [12] extended their work to the capacitated case. Finding a matching that is robust with respect to small modifications of the preferences have been studied by Chen et al [7].…”

“…We concentrate on the problem where to goal is to bribe the agents in order to make a given outcome stable. We consider a more general framework than the one in [5] and [12]. In our framework, the preferences are given with real numbers, and instead of swap distance we work with the 1 -norm.…”