Solving concisely expressed combinatorial auction problems

Combinatorial auctions provide a valuable mecha-nism for the allocation of goods in settings where buyer valuations exhibit complex structure with re-spect to substitutability and complementarity. Most algorithms are designed to work with explicit “flat” bids for concrete bundles of goods. However, logi-cal bidding languages allow the expression of com-plex utility functions in a natural and concise way, and have recently attracted considerable attention. Despite the power of logical languages, no current winner determination algorithms exploit the spe-cific structure of logically specified bids to solve problems more effectively. In this paper, we de-scribe techniques to do just this. Specifically, we propose a direct integer program (IP) formulation of the winner determination problem for bids in the LGB logical language. This formulation is linear in the size of the problem and can be solved effectively using standard optimization packages. We compare this formulation and its solution time to those of the corresponding set of flat bids, demonstrating the im-mense utility of exploiting the structure of logically expressed bids. We also consider an extension of LGB and show that these can also be solved using linear constraints.