Colorful linear programming, Nash equilibrium , and pivots

The colorful Carathéodory theorem, proved by Barany in 1982, states that given d+1 sets of points S_1,...,S_{d+1} in R^d, such that each S_i contains 0 in its convex hull, there exists a set subset T in the union of the S_i containing 0 in its convex hull and such that T intersects each S_i at most once. An intriguing question - still open - is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, Barany 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, and there are also other problems. We present new complexity results for colorful linear programming problems and propose a variant of the "Barany-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 applications 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

Tropical Carathéodory with Matroids

Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth

Tropical complementarity problems and Nash equilibria

Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time

Semigroups of Polyhedral Lattice Points: Convexity, Combinatorics, and Algebra

This dissertation explores problems of convexity, combinatorics, and algebra associated with semigroups of polyhedral lattice points.In \Cref{ChapterColored}, we first attempt to generalize and extend three well-known convexity theorems, including Helly theorem, Tverberg theorem, and Colorful Carath\'eodory theorem, to affine semigroups. We define a novel notion, chromatic representations of semigroup elements, this is in contrast to the colorful theory developed by B\'ar\'any et al. Later, we focus on one-dimensional affine semigroups, numerical semigroups, and study the number of chromatic solutions in numerical semigroups.In \Cref{ChapterWeighted}, we generalize the classical Hilbert functions and Hilbert series of a semigroup algebra to have weightings. We list three ways to add weightings, $q$-weighting, $r$-weighting, and $s$-weighting, and study their relationships. We find that the $q$-weighting can derive other weightings. Later, we specialize to the special family of semigroup algebras, the Ehrhart rings. We study and extend the properties of $h^{*}$-nonnegativity and Ehrhart–Macdonald reciprocity for the Ehrhart series under these three weightings.In \Cref{ChapterEhrhart}, we focus on the Ehrhart functions under the $s$-weighting and give a practical method to evaluate the $s$-weighted Ehrhart function. Specifically, we construct a new polytope, the weight-lifting polytope, and build a connection between the $s$-weighted Ehrhart function and the classical Ehrhart function. Later, we present several applications and experiments of our method in combinatorial representation theory and number theory.In \Cref{ChapterKakeya}, we discuss a long-standing conjecture, Kakeya's conjecture, which brings a surprising connection between numerical semigroups and symmetric polynomials. We give partial results, prove the conjecture for two variables, and outline a general computer proof for an arbitrary number of variables

Colorful linear programming, Nash equilibrium, and pivots

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