Hypergraph Valuations with Restricted Overlapping

In a single-item auction, each bidder submits a valuation for the good and the auctioneer typically awards the item, or good, to the highest bidder, thereby maximizing the social welfare. If multiple goods are auctioned simultaneously, not only does the task of maximizing social welfare (winner determination) become more complex, but eliciting bids becomes complex as well. Since the goods can exhibit substitute or complementary relationships, a valuation over the set G of goods may require exponential storage (at worst, a value for each subset of G). This problem motivates the study of bidding languages: fully or near-fully expressive languages for expressing valuations which aim to be concise for some useful valuation classes. One such bidding language uses hypergraphs to represent valuations; each good is represented by a node and each hyperedge (incident to an arbitrary number of vertices) represents the marginal benefit or loss from obtaining a particular bundle of items. This representation is fully expressive, so in order to solve the winner determination problem, we will need to restrict the graph in some way. While Abraham et al (2012) and Feige et al (2014) restrict the rank of the hypergraph—the maximal degree of any hyperedge—we instead restrict the degree to which hyperedges are allowed to overlap. We obtain polynomial-time winner determination algorithms for small amounts of overlapping.