The computational complexity of dynamic programming algorithm for combinatorial auctions

Auctions are important mechanisms for resource and task allocation in multi-agent systems. Combinatorial auctions where bidders can bid on bundles of items can lead to more economical revenue. This is a winner determination problem. But determining the winners is NP-complete. Rothkopf et al. (1998) presented a dynamic programming algorithm for this problem. It is an optimal algorithm found so far for the general case. In this paper we analyze its computational complexity exactly, prove that the computational complexity of the algorithm is Θ(3ⁿ) for n items. Compared with the result O(3ⁿ) of Rothkopf et al., it is more exact. This will help research on the approximate algorithm for the winner determination problem in combinatorial auctions, and for the optimal coalition structure generation in multi-agent systems.