相关论文: Extended Formulations for Sparsity Matroids
Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these…
Compressive sensing has become a powerful addition to uncertainty quantification when only limited data is available. In this paper we provide a general framework to enhance the sparsity of the representation of uncertainty in the form of…
We give explicit recursive constructions for the polytope of all matroids $\Omega_{r,n}$ in ranks 2 and 3 for all ground set sizes. This polytope was introduced in recent work by Ferroni and Fink as a tool for checking positivity…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
The Tutte polynomial is a crucial invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this…
The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already…
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as…
We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $\lambda^{s_n}$. In particular, we obtain a result which is…
For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the…
A subset $S$ of $\mathbb R^d$ has the Borsuk property if it can be decomposed into at most $d+1$ parts of diameter smaller than $S$. This is an important geometric property, inspired by a conjecture of Borsuk from the 1930s, which has…
A long-standing sample compression conjecture asks to linearly bound the size of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC) dimension of an arbitrary class. In this paper, we explore the rich metric and…
There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space.…
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove…
The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way.…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
We establish a sparsity in terms of $\ell_p$-summability and weighted $\ell_2$-summability for the coefficients of the Laguerre generalized piecewise-polynomial chaos expansion of solutions to parametric elliptic PDEs with log-Laplace…
Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…
We prove a KKM-type theorem for matroid colored families of set coverings of a polytope. This generalizes Gale's colorful KKM theorem as well as recent sparse-colorful variants by Sober\'on, and McGinnis and Zerbib.
We propose an algebraic combinatorial method for solving large sparse linear systems of equations locally - that is, a method which can compute single evaluations of the signal without computing the whole signal. The method scales only in…
Vertex Subset Problems (VSPs) are a class of combinatorial optimization problems on graphs where the goal is to find a subset of vertices satisfying a predefined condition. Two prominent approaches for solving VSPs are dynamic programming…