Combinatorial auctions provide a valuable mechanism for the allocation of goods in settings where buyer valuations exhibit complex structure with respect to substitutability and complementarity. Most algorithms are designed to work with explicit "flat" bids for concrete bundles of goods. However, logical bidding languages allow the expression of complex 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 specific structure of logically specified bids to solve problems more effectively. In this paper, we describe 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 immense 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.