Preference-constrained Oriented Matching

We introduce and study a combinatorial problem called preference-constrained oriented matching. This problem is defined on a directed graph in which each node has preferences over its out-neighbors, and the goal is to find a maximum-size matching on this graph that satisfies a certain preference constraint. One of our main results is a structural theorem showing that if the given graph is complete, then for any preference ordering there always exists a feasible matching that covers a constant fraction of the nodes. This result allows us to correct an error in a proof by Azar, Jain, and Mirrokni [1], establishing a lower bound on the price of anarchy in coordination mechanisms for scheduling. We also show that the preference-constrained oriented matching problem is APX-hard and give a constant-factor approximation algorithm for it.