Improving the Computational Efficiency of Combinatorial Auction Algorithms

This paper introduces two algorithms (both actually are variants of a single algorithm)—the First price Upper Bound, FUB, and Second price Lower Bound, SeLB, algorithms—aimed at serving as subroutines in computationally efficient algorithms for combinatorial auctions. The type of combinatorial auctions treated in the paper are the once where bids are placed on one unit of one or more items at a certain valuation of the combinatoin. Thus, he problem is to find a combination of bids that maximizes the total surplus, while the combination is feasible (i.e. at most one unit per commodity can be requested in the combination). Consequently, this problem is equivalent to weighted k-set packing. The FUB algorithm gives an upper bound on the total surplus. The SeLB algorithm gives 1) an approximate solution (and hence a lower bound) and 2) a ranking of bids, named the Second Price Surplus (SePS) rank. Empirical studies suggests that these algorithms are at least one to two orders of magnitudes faster than current algorithms for optimal winner determination, for example the algorithms by Sandholm [1] and Fujishima et al. [2]. The studies also show that the approximate solution often is of very high quality (often within 5% from the optimal solution). The ranking tends to rank bids such that bids with a high probability have a high rank—often less than 1% of the search space needs to be considered in order to find the optimal solution, and combinations with high surplus are found very early. The main implication of this is that it seems that FUB and SeLB algorithms can be run before the current algorithms in order to drastically improve the anytime performance (i.e. very rapidly come up with a high quality solution) and significantly reduce the time required for finding the optimal solution. Novel algorithms for winner determination based on repeated usage of FUB and SeLB are currently under evaluation.