A primal-dual approach for optimizing Ordered Weighted Average in perfect matching problems
Résumé
A perfect matching [Edmonds, 1965] for a (non necessarily bipartite) graph is a subset of non-adjacent edges that cover every vertex. In this work, we assume the graph is weighted and our aim is to find a perfect matching whose edge weights are fairly distributed. This issue is important for instance if one wants to be fair to the pairs of nodes in a matching. The standard formulation of the weighted perfect matching problem, which consists in finding a perfect matching that optimizes the sum of the edge weights, does not allow any control on the distribution of the edge weights. To model fairness, we use the Ordered Weighted Average (OWA) with decreasing OWA weights [Weymark, 1981], instead of the sum, to aggregate the edge weights. While OWA is a non-linear aggregating function, a 0,1-linear program can be formulated for finding an OWA optimal perfect matching, exploiting a linearization of OWA proposed by Ogryczak and Sliwinski (2003). For solving this problem, we propose an approximation scheme with a guaranteed performance based on a primal-dual approach.