Interval Subset Sum and Uniform-Price Auction Clearing

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, kmax]. 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 kISSPand solved accordingly.