On the Hardness of Welfare Maximization in Combinatorial Auctions with Submodular Valuations

We present a new type of monotone submodular functions: multi-peak submodular functions. Roughly speaking, given a family of sets F, we construct a monotone submodular function f with a high value f(S) for every set S ∈ F (a “peak”), and a low value on every set that does not intersect significantly any set in F. We use this construction to show that a better than (1 − 1 2e )-approximation ( 0.816) for welfare maximization in combinatorial auctions with submodular valuations is (1) impossible in the communication model, (2) NP-hard in the computational model where valuations are given explicitly. Establishing a constant approximation hardness for this problem in the communication model was a long-standing open question. The valuations we construct for the hardness result in the computational model depend only on a constant number of items, and hence the result holds even if the players can answer arbitrary queries about their valuation, including demand queries. We also study...

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Shahar Dobzinski, Jan Vondrák
• Keywords: Combinatorial Auctions, Communication Model, CORR 2012, Education, Submodular Functions