ILP-Based Heuristics for the Multi-Modal Stable Matching Problem

In this paper, we investigate the stable matching problem with multiple preferences in bipartite graphs, where each agent has various preference lists for all available partners with respect to different criteria. The problem requires that each matched agent must have exactly one partner and the obtained matching should be stable for all criteria. As our main contribution, we present an integer linear programming (ILP) model for determining whether there exists a globally stable matching in bipartite graphs, which has been proved to be NP-hard. Since the time consumed for solving ILPs might dramatically increase as the size of instances grows, we develop a preprocessing technique that helps to eliminate pairs that will never be a member of any globally stable matching and thus accelerates the computing process. We perform experiments on randomly generated preference lists and observe a significant speedup when we preprocess the instance before solving the ILPs. As there does not need to exist a perfect matching that is stable for all given criteria, we extend our ILP to the optimized version of the aforementioned problem, which asks to find a matching with maximum cardinality that is stable among all matched agents.