Improved Multi-unit Auction Clearing Algorithms with Interval (Multiple-Choice) Knapsack Problems

We study the interval knapsack problem (I-KP), and the interval multiple-choice knapsack problem (I-MCKP), as generalizations of the classic 0/1 knapsack problem (KP) and the multiple-choice knapsack problem (MCKP), respectively. Compared to singleton items in KP and MCKP, each item i in I-KP and I-MCKP is represented by a ([ai, bi], pi) pair, where integer interval [ai, bi] specifies the possible range of units, and pi is the unit-price. Our main results are a FPTAS for I-KP with time O(n log n + n/ǫ) and a FPTAS for I-MCKP with time O(nm/ǫ), and pseudo-polynomial-time algorithms for both I-KP and I-MCKP with time O(nM) and space O(n+M). Here n, m, and M denote number of items, number of item sets, and knapsack capacity respectively. We also present a 2-approximation of I-KP and a 3-approximation of I-MCKP both in linear time. We apply I-KP and I-MCKP to the single-good multi-unit sealedbid auction clearing problem where M identical units of a single good are auctioned. We focus on two bidding models, among them the interval model allows each bid to specify an interval range of units, and XOR-interval model allows a bidder to specify a set of mutually exclusive interval bids. The interval and XOR-interval bidding models correspond to I-KP and I-MCKP respectively, thus are solved accordingly. We also show how to compute VCG payments to all the bidders with an overhead of O(log n) factor. Our results for XOR-interval bidding model imply improved algorithms for the piecewise constant bidding model studied by Kothari et al. [18], improving their algorithms by a factor of Ω(n).