The Complexity of Determining the Uniqueness of Tarski’s Fixed Point under the Lexicographic Ordering
 Chuangyin Dang,
 Yinyu Ye
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Abstract
The wellknown Tarski’s fixed point theorem asserts that an increasing mapping from a complete lattice into itself has a fixed point. This theorem plays an important role in the development of supermodular games for economic analysis. Let C be a finite lattice consisting of all integer points in an ndimensional box and f be an increasing mapping from C into itself in terms of lexicographic ordering. It has been shown in the literature that, when f is given as an oracle, a fixed point of f can be found in polynomial time. The problem we consider in this paper is the complexity of determining whether or not f has a unique fixed point. We present a polynomialtime reduction of integer programming to an increasing mapping from C into itself. As a result of this reduction, we prove that, when f is given as an oracle, determining whether or not f has a unique fixed point is CoNP hard.
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 Title
 The Complexity of Determining the Uniqueness of Tarski’s Fixed Point under the Lexicographic Ordering
 Book Title
 Internet and Network Economics
 Book Subtitle
 6th International Workshop, WINE 2010, Stanford, CA, USA, December 1317, 2010. Proceedings
 Pages
 pp 455461
 Copyright
 2010
 DOI
 10.1007/9783642175725_38
 Print ISBN
 9783642175718
 Online ISBN
 9783642175725
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 6484
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Lexicographic Ordering
 Lattice
 Finite Lattice
 Increasing Mapping
 Fixed Point
 Integer Programming
 CoNP Completeness
 CoNP Hardness
 Industry Sectors
 eBook Packages
 Editors

 Amin Saberi ^{(16)}
 Editor Affiliations

 16. Department of Management Science and Engineering, Stanford University
 Authors

 Chuangyin Dang ^{(17)}
 Yinyu Ye ^{(18)}
 Author Affiliations

 17. Dept. of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Kowloon, Hong Kong SAR, China
 18. Dept. of Management Science & Engineering, Stanford University, Stanford, CA, 943054026
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