Linear Programming helps solving large multi-unit combinatorial auctions

TL;DR

This paper demonstrates that using Linear Programming to bound solutions in large multi-unit combinatorial auctions significantly improves pruning efficiency, enabling practical solutions for auctions with hundreds of goods and thousands of bids.

Contribution

The authors show that LP-based bounding is effective and practical for large auctions, introducing optimized heuristics for LP calls and bid ordering that outperform previous methods.

Findings

LP bounds enable extensive pruning in auction algorithms

Heuristics based on bid size and LP solutions improve efficiency

Large auctions with hundreds of goods can be solved practically

Abstract

Previous works suggested the use of Branch and Bound techniques for finding the optimal allocation in (multi-unit) combinatorial auctions. They remarked that Linear Programming could provide a good upper-bound to the optimal allocation, but they went on using lighter and less tight upper-bound heuristics, on the ground that LP was too time-consuming to be used repetitively to solve large combinatorial auctions. We present the results of extensive experiments solving large (multi-unit) combinatorial auctions generated according to distributions proposed by different researchers. Our surprising conclusion is that Linear Programming is worth using. Investing almost all of one's computing time in using LP to bound from above the value of the optimal solution in order to prune aggressively pays off. We present a way to save on the number of calls to the LP routine and experimental results comparing different heuristics for choosing the bid to be considered next. Those results show that the ordering based on the square root of the size of the bids that was shown to be theoretically optimal in a previous paper by the authors performs surprisingly better than others in practice. Choosing to deal first with the bid with largest coefficient (typically 1) in the optimal solution of the relaxed LP problem, is also a good choice. The gap between the lower bound provided by greedy heuristics and the upper bound provided by LP is typically small and pruning is therefore extensive. For most distributions, auctions of a few hundred goods among a few thousand bids can be solved in practice. All experiments were run on a PC under Matlab.