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Combinatorial Auctions$
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Peter Cramton, Yoav Shoham, and Richard Steinberg

Print publication date: 2005

Print ISBN-13: 9780262033428

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262033428.001.0001

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PRINTED FROM MIT PRESS SCHOLARSHIP ONLINE (www.mitpress.universitypressscholarship.com). (c) Copyright The MIT Press, 2022. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in MITSO for personal use.date: 07 July 2022

The Winner Determination Problem

The Winner Determination Problem

(p.297) 12 The Winner Determination Problem
Combinatorial Auctions

Daniel Lehmann

Rudolf Müller

Tuomas Sandholm

The MIT Press

This chapter defines and formulates a combinatorial optimization problem, called the winner determination problem, and examines its complexity properties. A range of alternative mathematical programming models including integer linear programming, weighted stable set in graphs, knapsack, and matching that covers variants of the problem related to particular bidding languages is also discussed. The chapter further highlights inapproximability results, and reviews the approximation algorithm, which is a polynomial time algorithm. It concludes that the problem, which is represented as an integer program, is NP-complete and can be tackled by applying three fundamentally different approaches, which it discusses.

Keywords:   winner determination problem, mathematical programming models, integer linear programming, bidding languages, approximation algorithms, polynomial time algorithm

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