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 roomm ates problem, which has a feasible solution if and only if the underlying r oommates problem has a stable matching. Furthermore, for certain special we ight functions on the edges, we construct a a-approximation algorithm for t he optimal stable roommates problem. Our technique exploits features of the geometry of fractional solutions of this formulation. For the stable marri age problem, we show that a related geometry allows us to express any fract ional solution in the stable marriage polytope as a convex combination of s table marriage solutions. This also leads to a genuinely simple proof of th e integrality of the stable marriage polytope.