The Complexity of Determining the Uniqueness of Tarski’s Fixed Point under the Lexicographic Ordering

  • Chuangyin Dang
  • Yinyu Ye
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)

Abstract

The well-known 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 n-dimensional 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 polynomial-time 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 Co-NP hard.

Keywords

Lexicographic Ordering Lattice Finite Lattice Increasing Mapping Fixed Point Integer Programming Co-NP Completeness Co-NP Hardness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Chuangyin Dang
    • 1
  • Yinyu Ye
    • 2
  1. 1.Dept. of Manufacturing Engineering & Engineering ManagementCity University of Hong KongKowloonHong Kong SAR, China
  2. 2.Dept. of Management Science & EngineeringStanford UniversityStanford

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