Accelerating Vickrey Payment Computation in Combinatorial Auctions for an Airline Alliance

Abstract

Among all the variations of general combinatorial auctions, the Vickrey auction is essentially the only incentive-compatible auction. Furthermore, it is individual rational and weakly budget-balanced. In many cases these properties are very desirable. However, computing the winners and their payments in a Vickrey auction involves solving several NP-complete problems. While there have been many approaches to solve the winner determination problem via search, this search has not been extended to compute the Vickrey payments. The naive approach is to consecutively solve each problem using the same search algorithm. We present an extension to this procedure to accelerate the computation of Vickrey payments using a simple backtrack algorithm. However, our results can be applied to sophisticated branch-and-bound solvers as well. We test our approach on data evolving from a Lufthansa flight schedule. Data of this type might be of interest, since authentic data for combinatorial auctions is rare and uch sought after. A remarkable result is that after solving the winner determination problem we can provide bounds for the remaining problems that differ from the optimal solution by only 2.2% on average. We as well manage to obtain a rapid speedup by tolerating small deviations from the optimal solutions. In all cases, the actual deviations are much smaller than the allowed deviations.

This work was partially supported by the German Science Foundation (DFG) priority programme 1126 Algorithmics of Large and Complex Netzworks, project Integration of network design and fleet assignment in airline optimization under grant MO 285/15-2, by the Future and Emerging Technologies programme of EU under EU Contract 001907 DELIS, Dynamically Evolving, Large Scale Information Systems, and by the Future and Emerging Technologies programme of EU under EU Contract 33116 FLAGS, Foundational Aspects of Global Computing Systems.