ISAAC 2006: Algorithms and Computation pp 494-506

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

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

## 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).

## Keywords

Knapsack Problem Combinatorial Auction Large Item Small Item Reverse Auction
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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