Maximum Stable Matching with Matroids and Partial Orders

The Stable Marriage problem (SM), solved by the famous deferred acceptance algorithm of Gale and Shapley (GS), has many natural generalizations. If we allow ties in preferences, then the problem of finding a maximum stable matching becomes NP-hard, and the best known approximation ratio is 1.5 (McDermid 2009, Paluch 2011, Z. Kir\'aly 2012), achievable by running GS on a cleverly constructed modified instance. Another elegant generalization of SM is the matroid kernel problem introduced by Fleiner (2001), which is solvable in polynomial time using an abstract matroidal version of GS. Our main result is a simple 1.5-approximation algorithm for the matroid kernel problem when preferences are given as interval orders -- a broad subclass of partial orders that covers many applications beyond preferences with ties. In addition, for the bipartite matching case, we show that the output of our algorithm also 1.5-approximates the LP-optimum of the relaxation of the corresponding Integer Program, which shows that the integrality gap is at most 1.5 for the interval order case. To contrast this with hardness results, we show that if arbitrary partial orders are allowed in the preferences, then even in the bipartite matching case, the problem becomes hard to approximate within a factor better than 2 assuming the Unique Games Conjecture, and the integrality gap becomes 2.