Winner Determination in Discount Auctions

Discount auctions is a market mechanism for buying heterogeneous items in a single auction. The bidders are suppliers and a bid consists of individual cost for each of the items and a non-decreasing discount function defined over the number of items. The winner determination problem faced by the buyer is to determine the winning suppliers and their corresponding winning items. We show that this problem is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal NP}$\end{document}-hard upon reduction from the set covering problem. The problem has an embedded network structure, which is exploited to develop heuristics and an exact branch and bound algorithm. Computational experiments were performed to evaluate the proposed algorithms.