Walrasian Equilibria in Markets with Small Demands
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We study the complexity of finding a Walrasian equilibrium in markets where the agents have k-demand valuations, where k is a constant. This means that the maximum value of every agent comes from a bundle of size at most k. Our results are threefold. For unit-demand agents, where the existence of a Walrasian equilibrium is guaranteed, we show that the problem is in quasi-NC. Put differently, we give the first parallel algorithm that finds a Walrasian equilibrium in polylogarithmic time. This comes in striking contrast to all existing algorithms that are highly sequential. For k=2, we show that it is NP-hard to decide if a Walrasian equilibrium exists even if the valuations are submodular, while for k=3 the hardness carries over to budget-additive valuations. In addition, we give a polynomial-time algorithm for markets with 2-demand single-minded valuations, or unit-demand valuations. Our last set of results consists of polynomial-time algorithms for k-demand valuations in unbalanced markets; markets where the number of items is significantly larger than the number of agents, or vice versa.
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