An investigation of representations of combinatorial auctions

Combinatorial auctions (CAs) are an important mechanism for allocating multiple goods while allowing self-interested agents to specify preferences over bundles of items. Winner determination for a CA is known to be NP-complete. However, restricting the problem can allow us to solve winner determination in polynomial time. These restrictions sometimes apply to the CA's representation. There are two commonly studied, and structurally different graph representations of a CA: bid graphs and item graphs. We study the relationship between these two representations. We show that for a given combinatorial auction, if a graph with maximum cycle length three is a valid item graph for the auction, then its bid graph representation is a chordal graph. Next, we present a new technique for constructing item graphs using a novel definition of equivalence among combinatorial auctions. The solution to the WDP for a given CA can easily be translated to a solution on an equivalent CA. We use our technique to simplify item graphs, and show that if a CA's bid graph is chordal, then there exists an equivalent CA with a valid item graph of treewidth one, for which a solution to the WDP is known to be efficient. This result demonstrates how CA equivalence can simplify the structure of item graphs and lead to more efficient solutions to the WDP, which are also a solutions to the WDP for the original auctions.