Colorful linear programming, Nash equilibrium, and pivots

Colorful linear programming, Nash equilibrium, and pivots The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d+1 sets of points S1,…,Sd+1 in Rd, with each Si containing 0 in its convex hull, there exists a set T⊆⋃i=1d+1Si containing 0 in its convex hull and such that |T∩Si|≤1 for all i∈{1,…,d+1}. An intriguing question–still open–is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming.The traditional applications of colorful linear programming lie in discrete geometry. In this paper, we study its relations with other areas, such as game theory, operations research, and combinatorics. Regarding game theory, we prove that computing a Nash equilibrium in a bimatrix game is a colorful linear programming problem. We also formulate an optimization problem for colorful linear programming and show that as for usual linear programming, deciding and optimizing are computationally equivalent. We discuss then a colorful version of Dantzig’s diet problem. We also propose a variant of the Bárány algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm and we discuss its computational efficiency. Related complexity and combinatorial results are also provided. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Applied Mathematics Elsevier

Colorful linear programming, Nash equilibrium, and pivots

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Publisher
North-Holland
Copyright
Copyright © 2016 Elsevier B.V.
ISSN
0166-218X
D.O.I.
10.1016/j.dam.2016.10.006
Publisher site
See Article on Publisher Site

Abstract

The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d+1 sets of points S1,…,Sd+1 in Rd, with each Si containing 0 in its convex hull, there exists a set T⊆⋃i=1d+1Si containing 0 in its convex hull and such that |T∩Si|≤1 for all i∈{1,…,d+1}. An intriguing question–still open–is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming.The traditional applications of colorful linear programming lie in discrete geometry. In this paper, we study its relations with other areas, such as game theory, operations research, and combinatorics. Regarding game theory, we prove that computing a Nash equilibrium in a bimatrix game is a colorful linear programming problem. We also formulate an optimization problem for colorful linear programming and show that as for usual linear programming, deciding and optimizing are computationally equivalent. We discuss then a colorful version of Dantzig’s diet problem. We also propose a variant of the Bárány algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm and we discuss its computational efficiency. Related complexity and combinatorial results are also provided.

Journal

Discrete Applied MathematicsElsevier

Published: May 11, 2018

References

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