Stable matchings and linear programming

Stable matchings and linear programming

THE stable matching problem was introduced by Gale and Shapley as a model of how to assign students to colleges. It is a classic that has inspired a flood of papers exploring generalizations, variations and applications. For a comprehensive account of these, see Roth and Sotomayor as well as Roth. A stable matching is a matching in a bipartite graph that satisfies additional conditions. Just as...

Stable Matchings

Limitations of Linear Programming Techniques for Bounded Color Matchings

Given a weighted graph G = (V,E,w), with weight function w : E → Q, a matching M is a set of pairwise non-adjacent edges. In the optimization setting, one seeks to find a matching of maximum weight. In the multi-criteria (or multi-budgeted) setting, we are also given ` length functions α1, . . . , α` assigning positive numbers to each edge and ` numbers β1, . . . , β` ∈ Q, each one associated w...

Internally Stable Matchings and Exchanges

Stability is a central concept in exchange-based mechanism design. It imposes a fundamental requirement that no subset of agents could beneficially deviate from the outcome prescribed by the mechanism. However, deployment of stability in an exchange mechanism presents at least two challenges. First, it reduces social welfare and sometimes prevents the mechanism from producing a solution. Second...

Stable Schedule Matchings

In order to treat a natural schedule matching problem related with worker-firm matchings, we generalize some theorems of Baiou–Balinski and Alkan–Gale by applying a fixed point method of Fleiner.