Optimal allocation problem with quadratic utility functions and its relationship with graph cut problem

Abstract

We discuss the optimal allocation problem in combinatorial auctions, where the items are allocated to bidders so that the sum of the bidders' utilities is maximized. In this paper, we consider the case where utility functions are given by quadratic functions; the class of such utility functions has a succinct representation but is sufficiently general. The main aini of this paper is to show the computational complexity of the optimal allocation problem with quadratic utility functions. We consider the cases where utility functions are submodular and supermodular, and show NP-hardness and/or polynomial-time cxact/approximation algorithms. These results are given by using the relationship with graph cut problems such as the mm/max cut problem and the multiway cut problem. © The Operations Research Society of Japan.