Decomposing combinatorial auctions and set packing problems

Combinatorial auctions allow bidders to bid on bundles of items rather than just on single items. The winner determination problem in combinatorial auctions is the problem of determining the allocation of items to bidders such that the sum of the accepted bid prices is maximized. This problem is equivalent to the wellknown maximum-weight set packing problem. Even though these problems are NP-hard in general, they can be solved in polynomial time on instances whose associated item graphs have bounded treewidth (called structured item graphs). However, the tractability of determining whether for a given problem instance a structured item graph of fixed treewidth exists (and if so, computing one efficiently) was an open problem. In this article, we solve this problem by proving that deciding the existence of structured item graphs is computationally intractable, even for treewidth 3. Motivated by this unfavorable complexity result, we investigate other structural restrictions, and we show that the notion of hypertree decomposition, a wellstudied measure of hypergraph cyclicity, turns out to be most useful here. Indeed, we show that the winner determination problem is solvable in polynomial time on instances whose dual auction hypergraphs have bounded hypertree width. Our solution method is based on encoding winner determination via a constraint satisfaction optimization problem and on exhibiting an algorithm to solve this latter problem efficiently for such structurally restricted instances. The class of tractable instances identified by our approach, while being efficiently recognizable, properly contains the class of instances having a structured item graph. Moreover, on the larger class, our method solves winner determination with the same asymptotic complexity as the best algorithm proposed in the literature for the subclass of structured item graphs. Hypertree decompositions can equally profitably be applied to the maximum-weight independent set problem, which is the dual problem of maximum-weight set packing.