1,940 research outputs found
A Tutte polynomial inequality for lattice path matroids
Let be a matroid without loops or coloops and let be its Tutte
polynomial. In 1999 Merino and Welsh conjectured that holds for graphic matroids. Ten years later, Conde and
Merino proposed a multiplicative version of the conjecture which implies the
original one. In this paper we prove the multiplicative conjecture for the
family of lattice path matroids (generalizing earlier results on uniform and
Catalan matroids). In order to do this, we introduce and study particular
lattice path matroids, called snakes, used as building bricks to indeed
establish a strengthening of the multiplicative conjecture as well as a
complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction
Unavoidable parallel minors of regular matroids
This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency
(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing
Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.
The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.
Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]
Lattice path matroids: the excluded minors
A lattice path matroid is a transversal matroid for which some collection of
incomparable intervals in some linear order on the ground set is a
presentation. We characterize the minor-closed class of lattice path matroids
by its excluded minors.Comment: 13 pages, 2 figure
Polynomials with the half-plane property and matroid theory
A polynomial f is said to have the half-plane property if there is an open
half-plane H, whose boundary contains the origin, such that f is non-zero
whenever all the variables are in H. This paper answers several open questions
regarding multivariate polynomials with the half-plane property and matroid
theory.
* We prove that the support of a multivariate polynomial with the half-plane
property is a jump system. This answers an open question posed by Choe, Oxley,
Sokal and Wagner and generalizes their recent result claiming that the same is
true whenever the polynomial is also homogeneous.
* We characterize multivariate multi-affine polynomial with real coefficients
that have the half-plane property (with respect to the upper half-plane) in
terms of inequalities. This is used to answer two open questions posed by Choe
and Wagner regarding strongly Rayleigh matroids.
* We prove that the Fano matroid is not the support of a polynomial with the
half-plane property. This is the first instance of a matroid which does not
appear as the support of a polynomial with the half-plane property and answers
a question posed by Choe et al.
We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat
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