Abstract. Given a bipartite graph G = (A ∪ B, E) with strict preference lists and e * ∈ E, we ask if there exists a popular matching in G that contains the edge e * . We call this the popular edge problem. A matching M is popular if there is no matching M such that the vertices that prefer M to M outnumber those that prefer M to M . It is known that every stable matching is popular; however G may have no stable matching with the edge e * in it. In this paper we identify another natural subclass of popular matchings called "dominant matchings" and show that if there is a popular matching that contains the edge e * , then there is either a stable matching that contains e * or a dominant matching that contains e * . This allows us to design a linear time algorithm for the popular edge problem. We also use dominant matchings to efficiently test if every popular matching in G is stable or not.

No abstract

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (A∪ P, E) with weights on the edges in E, and with lower and upper quotas on the vertices in P.

An unceasing problem of our prevailing society is the fair division of goods. The problem of proportional cake cutting focuses on dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share. In this article, we investigate the problem of proportional division with unequal shares, where each player is entitled to receive a predetermined portion of the cake. Our main contribution is threefold. First we present a protocol for integer demands, which delivers a proportional solution in fewer queries than all known protocols. By giving a matching lower bound, we then show that our protocol is asymptotically the fastest possible. Finally, we turn to irrational demands and solve the proportional cake cutting problem by reducing it to the same problem with integer demands only. All results remain valid in a highly general cake cutting model, which can be of independent interest.