ON ALGORITHMS FOR THE COLOURFUL LINEAR PROGRAMMING FEASIBILITY PROBLEM

Abstract

<p>Given colourful sets S_1,..., S_{d+1} of points in R^d and a point p in R^d, the colourful linear programming problem is to express p as a convex combination of points x_1,...,x_{d+1} with x_i in S_i for each i. This problem was presented by Bárány and Onn in 1997, it is still not known if a polynomial-time algorithm for the problem exists. The monochrome version of this problem, expressing p as a convex combination of points in a set S, is a traditional linear programming feasibility problem. The colourful Carathéodory Theorem, due to Bárány in 1982, provides a sufficient condition for the existence of a colourful set of points containing p in its convex hull. Bárány's result was generalized by Holmsen et al. in 2008 and by Arocha et al. in 2009 before being recently further generalized by Meunier and Deza. We study algorithms for colourful linear programming under the conditions of Bárány and their generalizations. In particular, we implement the Meunier-Deza algorithm and enhance previously used random case generators. Computational benchmarking and a performance analysis including a comparison between the two algorithms of Bárány and Onn and the one of Meunier and Deza, and random picking are presented.</p>