Rank three matroids are Rayleigh

A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities.Comment: 11 pages, 3 figures, 3 table

Quaternionic Geometry of Matroids

Building on a recent joint paper with Sturmfels, here we argue that the combinatorics of matroids is intimately related to the geometry and topology of toric hyperkaehler varieties. We show that just like toric varieties occupy a central role in Stanley's proof for the necessity of McMullen's conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkaehler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkaehler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari, leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.Comment: 11 page

Polymatroid Prophet Inequalities

Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen. The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint. In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing schemePeer ReviewedPostprint (author's final draft

Finiteness theorems for matroid complexes with prescribed topology

It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$ for any natural number $k$. In this paper we study the question at the other end of the $h$-vector: Are there only finitely many $(d-1)$-dimensional simplicial complexes with $h_d=k$ for any given $k$? The answer is no if we consider general complexes, but when focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. We prove the answer is yes in cases (i) and (iii) and conjecture it is also true in case (ii).Comment: to appear in European Journal of Combinatoric