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

  • Yunhong Zhou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)


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 ([a i , b i ], p i ) pair, where integer interval [a i , b i ] specifies the possible range of units, and p i is the unit-price. Our main results are a FPTAS for I-KP with time O(n logn + n/ε 2) and a FPTAS for I-MCKP with time O(nm /ε), 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 sealed-bid 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(logn) 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).


Knapsack Problem Combinatorial Auction Large Item Small Item Reverse Auction 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yunhong Zhou
    • 1
  1. 1.HP LabsPalo AltoUSA

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