The Geometry of Fractional Stable Matchings and Its Applications

We study the classical stable marriage and stable roommates problems using a polyhedral approach. We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching. Furthermore, for certain special weight functions on the edges, we construct a 2-approximation algorithm for the optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions. This also leads to a genuinely simple proof of the integrality of the stable marriage polytope.