Interval subset sum and uniform-price auction clearing

Abstract

We study the interval subset sum problem (ISSP), a generalization of the classic subset-sum problem, where given a set of intervals, the goal is to choose a set of integers, at most one from each interval, whose sum best approximates a target integer T. For the cardinality constrained interval subset-sum problem (kISSP), at least kmin and at most kmax integers must be selected. Our main result is a fully polynomial time approximation scheme for ISSP, with time and space both O(n · 1/ε). For kISSP, we present a 2-approximation with time O(n), and a FPTAS with time O(n · kmax · 1/ε). Our work is motivated by auction clearing for uniform-price multi-unit auctions, which are increasingly used by security firms to allocate IPO shares, by governments to sell treasury bills, and by corporations to procure a large quantity of goods. These auctions use the uniform price rule - the bids are used to determine who wins, but all winning bidders receive the same price. For procurement auctions, a firm may even limit the number of winning suppliers to the range [kmin, k max]. We reduce the auction clearing problem to ISSP, and use approximation schemes for ISSP to solve the original problem. The cardinality constrained auction clearing problem is reduced to kISSP and solved accordingly. © Springer-Verlag Berlin Heidelberg 2005.