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 R , such that each Si contains 0 in its convex hull, there exists a set T ⊆ ⋃ d+1 i=1 Si 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. We present new complexity results for colorful linear programming problems and propose a variant of the “Bárány-Onn” 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. Some combinatorial versions of the colorful Carathéodory theorem are also discussed from an algorithmic point of view. Finally, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner’s lemma.