We study the pure exploration problem subject to a matroid constraint (Best-Basis) in a stochastic multi-armed bandit game. In a Best-Basis instance, we are given n stochastic arms with unknown reward distributions, as well as a matroid M over the arms. Let the weight of an arm be the mean of its reward distribution. Our goal is to identify a basis of M with the maximum total weight, using as f...

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simpli...

Cycle bases for lattices of binary matroids with no Fano dual minor and their one-element extensions Abstract In this paper we study the question of existence of a cycle basis (that is, a basis consisting only of cycles) for the lattice Z(M) generated by the cycles of a binary matroid M. We show that, if M has no Fano dual minor, then the set of fundamental circuits of any maximal independent s...

A cosimple regular matroid M does not have disjoint circuits if and only ifM ∈ {M(K3,3),M(Kn) (n ≥ 3)}. This extends a former result of Erdös and Pósa on graphs without disjoint circuits.

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field k. If a matroid M is isomorphic to an algebraic matroid the latter is called an algebraic representation of M. Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in...

Acknowledgments The support of many caring people made completing this dissertation possible. First, thanks to my wife Paula for her love and understanding through the many ups and downs. Thanks to my family and friends who cheered me on through these many years in school. Thanks to Linda Gleason for showing by her example how to care for students and love mathematics. Thanks to James Oxley for...

Fix a matroid N . A matroid M is N -fragile if, for each element e of M , at least one of M\e and M/e has no N -minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N -minor and are N -fragile have branch width bounded by a constant depending only on q and N . A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class w...

As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph G = (V,E), a multiset R = {r1, . . . , rt} of vertices in V , and a matroid M on R. We prove a necessary and sufficient condition for G to be decomposed into t edge-disjoint subgraphs G1 = ...

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the OrlikSolomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, call...

In this paper, some characterizations of fuzzifying strong compactness are given, including in terms nets and presubbases. Several locally the framework topology introduced mapping theorems obtained.