A Branch-and-Price Algorithm and New Test Problems for Spectrum Auctions

When combinatorial bidding is permitted in auctions, such as the proposed FCC auction #31, the resulting full valuations and winner-determination problem can be computationally challenging. We present a branch-and-price algorithm based on a set-packing formulation originally proposed by Dietrich and Forrest (2002). This formulation has a variable for every possible combination of winning bids for each bidder. Our algorithm exploits the structure of the XOR-of-OR bidding-language used by the FCC. We also present a new methodology to produce realistic test problems based on the round-by-round-results of FCC auction #4. We generate 2639 test problems which involve 99 items and are substantially larger than most of the previously used benchmark problems. Since there are no real-life test problems for combinatorial spectrum auctions with the XOR-of-OR language, we used these test problems to observe the computational behavior of our algorithm. Our algorithm can solve all but one test problem within ten minutes, appears to be very robust and compares for difficult instances favorably to the natural formulation solved using a commercial optimization package with default settings. Although spectrum auctions are used as the guiding example to describe the merits of branchand-price for combinatorial auctions our approach applies to auctions of multiple goods in other scenarios similarly.