Bipartite Envy-Free Matching
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Bipartite Envy-Free Matching (BEFM) is a relaxation of perfect matching. In a bipartite graph with parts X and Y, a BEFM is a matching of some vertices in X to some vertices in Y, such that each unmatched vertex in X is not adjacent to any matched vertex in Y (so the unmatched vertices do not "envy" the matched ones). The empty matching is always a BEFM. This paper presents sufficient and necessary conditions for the existence of a non-empty BEFM. These conditions are based on cardinality of neighbor-sets, similarly to Hall's condition for the existence of a perfect matching. The conditions can be verified in polynomial time, and in case they are satisfied, a non-empty BEFM can be found by a polynomial-time algorithm. The paper presents some applications of BEFM as a subroutine in fair division algorithms.
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