On Matrices over a Polynomial Ring with Restricted Subdeterminants

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We investigate in particular matrices whose subdeterminants all lie in a fixed set $S\subseteq\mathbb{Z}[x]$. Such matrices, which we call totally $S$-modular matrices, are closed with respect to taking submatrices, so it is natural to look at minimally non-totally $S$-modular matrices which we call forbidden minors for $S$. Among other results, we prove that if $S$ is finite, then the set of all determinants attained by a forbidden minor for $S$ is also finite. Specializing to the integers, we subsequently obtain the following positive complexity result: the recognition problem for totally $\pm\{0,1,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{-3,-2,1,2\}$ and the integer linear optimization problem for totally $\pm\{ 0,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{ -2,1\}$ can be solved in polynomial time

Sparsity and integrality gap transference bounds for integer programs

We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{cx: x in P, x integer}, where P={x: Ax=b, x nonnegative} is a polyhedron in the standard form determined by an integer mxn matrix A and an integer vector b. The main result of the paper shows that the integrality gap drops exponentially in the size of support of the optimal solutions that correspond to the vertices of the integer hull of the polyhedron P. Additionally, we obtain a new proximity bound that estimates the distance from any point of P to its nearest integer point in P. The proofs make use of the results from the geometry of numbers and convex geometry

Proximity and flatness bounds for linear integer optimization

We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. - Proximity bounds: Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists)? - Flatness bounds: If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, non-zero, normal vector that intersect the polyhedron? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane: if a polygon $K\subseteq\mathbb{R}^2$ satisfies $\tau K \subseteq K^{\circ}$, where $\tau$ denotes $90^{\circ}$ counterclockwise rotation and $K^{\circ}$ denotes the polar of $K$, then the area of $K^{\circ}$ is at least 3.Comment: This manuscripts greatly builds upon a related IPCO2022 paper. For all of the overlapping material, the current manuscript provides improved results. Additionally, the current manuscript derives new connections to flatness and generalizations to {0,k,2k} modular problem

Distance-sparsity transference for vertices of corner polyhedra

We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it gives an exponential (in the size of support of a solution) improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support

Distance-sparsity transference for vertices of corner polyhedra

We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it implies an exponential in size of support improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support

Proximity bounds for random integer programs

We study proximity bounds within a natural model of random integer programs of the type max cx : Ax = b, X œ Z >= 0, where A œ Z^mXn of rank m, b œ Z^m and c œ Z^n. We prove that (up to a constant depending on the dimension n) the proximity is generally bounded by ∆m(A)^1/(n-m), where ∆m(A) is the maximal absolute value of an m x m subdeterminant of A. This is significantly better than the best deterministic bounds which are linear in ∆m(A)

Lattice points, oriented matroids, and zonotopes

The first half of this dissertation concerns the following problem: Given a lattice in R^d which refines the integer lattice Z^d, what can be said about the distribution of the lattice points inside of the half-open unit cube [0,1)^d? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory. In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.Ph.D