Matroids over partial hyperstructures

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Pl\"ucker functions, and dual pairs, and establish some basic duality results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. For example, if $F$ is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong $F$-matroids coincide. We also give examples of tracts $F$ and weak $F$-matroids which are not strong. Our theory of matroids over tracts is closely related to, but more general than, "matroids over fuzzy rings" in the sense of Dress and Dress-Wenzel.Comment: 35 pages. v2: Final version to appear in Advances in Mathematics. Numerous minor updates and revisions. v1: This paper generalizes and subsumes the results of arXiv:1601.01204. Our treatment now includes partial fields, for example, in addition to hyperfield

Matroids over hyperfields

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.Comment: 31 pages. v2: Fixed a few errors, added some new examples and remarks, added Theorem 4.17, removed the "Brief chronology" from v1. v3: Fixed some minor errors, streamlined the exposition. v4: Fixed a major error and added a co-author; see Section 1.7 for further details. v5: Added section 5 on doubly distributive hyperfields, Example 3.31 (due to Daniel Weissauer), and Theorem 3.

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig

The Z-polynomial of a matroid

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the Z-polynomial in which the symmetry is a manifestation of Poincare duality

On the relation between hyperrings and fuzzy rings

We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields.Comment: 22 pages, v2: strengthened result and added coautho

On the Construction of Substitutes

Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Since gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability and fully describe all substitutes with at most 4 items

Determinantal probability measures

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made other small change

Interdicting Structured Combinatorial Optimization Problems with {0,1}-Objectives

Interdiction problems ask about the worst-case impact of a limited change to an underlying optimization problem. They are a natural way to measure the robustness of a system, or to identify its weakest spots. Interdiction problems have been studied for a wide variety of classical combinatorial optimization problems, including maximum $s$-$t$ flows, shortest $s$-$t$ paths, maximum weight matchings, minimum spanning trees, maximum stable sets, and graph connectivity. Most interdiction problems are NP-hard, and furthermore, even designing efficient approximation algorithms that allow for estimating the order of magnitude of a worst-case impact, has turned out to be very difficult. Not very surprisingly, the few known approximation algorithms are heavily tailored for specific problems. Inspired by an approach of Burch et al. (2003), we suggest a general method to obtain pseudoapproximations for many interdiction problems. More precisely, for any $\alpha>0$, our algorithm will return either a $(1+\alpha)$-approximation, or a solution that may overrun the interdiction budget by a factor of at most $1+\alpha^{-1}$ but is also at least as good as the optimal solution that respects the budget. Furthermore, our approach can handle submodular interdiction costs when the underlying problem is to find a maximum weight independent set in a matroid, as for example the maximum weight forest problem. The approach can sometimes be refined by exploiting additional structural properties of the underlying optimization problem to obtain stronger results. We demonstrate this by presenting a PTAS for interdicting $b$-stable sets in bipartite graphs

Isotropical Linear Spaces and Valuated Delta-Matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D

Realization spaces of matroids over hyperfields

We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler). More precisely, given a matroid M and a hyperfield H we determine the space of all H-matroids over M. This can be seen as the matroid stratum of the hyperfield Grassmannians in the sense of Anderson and Davis. We give different descriptions of these realization spaces (e.g., in terms of Tutte groups or projective classes), allowing for explicit computations. When the hyperfield at hand is topological, the realization spaces have a natural topology. In this case, our models carry the correct homeomorphism type. As applications of our methods we obtain a theorem on the existence of phased matroids that are not realizable over the complex field and are not chirotopal, as well as a result on the diffeomorphism type of complex hyperplane arrangements whose underlying matroid is uniform.Comment: 44 pages. Several typos and errors corrected, in particular in the statement of Theorem 4.4. Proof of Theorem 4.1 corrected and several proofs simplifie