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

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Algorithms and Computation (ISAAC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4288))

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Abstract

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).

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Authors and Affiliations

  1. HP Labs, 1501 Page Mill Rd, Palo Alto, CA, 94304, USA

    Yunhong Zhou

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  1. Yunhong Zhou
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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Japan

    Tetsuo Asano

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Zhou, Y. (2006). Improved Multi-unit Auction Clearing Algorithms with Interval (Multiple-Choice) Knapsack Problems. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_50

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  • DOI: https://doi.org/10.1007/11940128_50

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49694-6

  • Online ISBN: 978-3-540-49696-0

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