arXiv subjects

Marcel Celaya

Publications and source records attributed to Marcel Celaya.

11 recordsLinked to original sources

arXiv2026

Polynomial Matrices in Integer Programming With Restricted Subdeterminants

We introduce a framework for tackling questions in discrete optimization associated with parametric constraint matrices. More precisely, the constraint matrices have entries that are polynomials in one variable and all subdeterminants of these matrices are polynomials in a given prescribed set $S$. Two key problems arise in this context. The first is the recognition problem: can a matrix of this form be recognized in polynomial time? The second is the optimization problem: given an integer program whose constraint matrix is of this form, can it be solved in polynomial time? We answer both questions affirmatively for a particular set $S$ consisting of nine linear forms. The matrices we consider are of themselves independent interest; they arise as matrix projections of certain bimodular matrices that admit two distinct unimodular projections.

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arXiv2024

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.

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arXiv2024

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.

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arXiv2022

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 $τK \subseteq K^{\circ}$, where $τ$ denotes $90^{\circ}$ counterclockwise rotation and $K^{\circ}$ denotes the polar of $K$, then the area of $K^{\circ}$ is at least 3.

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arXiv2021

Improving the Cook et al. Proximity Bound Given Integral Valued Constraints

Consider a linear program of the form $\max\;c^{\top}x:Ax\leq b$, where $A$ is an $m\times n$ integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardos proved that, given an optimal solution $x^{*}$, if an optimal integral solution $z^{*}$ exists, then it may be chosen such that $\left\Vert x^{*}-z^{*}\right\Vert _{\infty}<nΔ$, where $Δ$ is the largest magnitude of any subdeterminant of $A$. Since then an open question has been to improve this bound, assuming that $b$ is integral valued too. In this manuscript we show that $nΔ$ can be replaced with $\frac{n}{2}\cdotΔ$ whenever $n\geq2$. We also show that, in certain circumstances, the factor $n$ can be removed entirely.

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arXiv2020

Oriented Matroids from Triangulations of Products of Simplices

We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.

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arXiv2020

Patchworking Oriented Matroids

In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological representation of the oriented matroid directly from the polyhedral structure of the triangulation, hence finding a combinatorial manifestation of patchworking besides tropical algebraic geometry. We achieve this by rephrasing the patchworking procedure as a controlled cell merging process, guided by the structure of tropical oriented matroids. A key insight is a new promising technique to show that the final cell complex is regular.

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arXiv2020

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.

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arXiv2017

On the span of lattice points in a parallelepiped

Let $Λ\subset\mathbf{R}^{n}$ be a lattice which contains the integer lattice $\mathbf{Z}^{n}$. We characterize the space of linear functions $\mathbf{R}^{n}\rightarrow\mathbf{R}$ which vanish on the lattice points of $Λ$ lying in the half-open unit cube $[0,1)^{n}$. We also find an explicit formula for the dimension of the linear span of $Λ\cap[0,1)^{n}$. The results in this paper generalize and are based on the Terminal Lemma of Reid, which is in turn based upon earlier work of Morrison and Stevens on the classification of four dimensional isolated Gorenstein terminal cyclic quotient singularities.

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arXiv2013

Morphic Words and Nested Recurrence Relations

We explore a family of nested recurrence relations with arbitrary levels of nesting, which have an interpretation in terms of fixed points of morphisms over a countably infinite alphabet. Recurrences in this family are related to a number of well-known sequences, including Hofstadter's G sequence and the Conolly and Tanny sequences. For a recurrence a(n) in this family with only finitely terms, we provide necessary and sufficient conditions for the limit a(n)/n to exist.

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arXiv2012

An Undecidable Nested Recurrence Relation

Roughly speaking, a recurrence relation is nested if it contains a subexpression of the form ... A(...A(...)...). Many nested recurrence relations occur in the literature, and determining their behavior seems to be quite difficult and highly dependent on their initial conditions. A nested recurrence relation A(n) is said to be undecidable if the following problem is undecidable: given a finite set of initial conditions for A(n), is the recurrence relation calculable? Here calculable means that for every n >= 0, either A(n) is an initial condition or the calculation of A(n) involves only invocations of A on arguments in {0,1,...,n-1}. We show that the recurrence relation A(n) = A(n-4-A(A(n-4)))+4A(A(n-4)) +A(2A(n-4-A(n-2))+A(n-2)). is undecidable by showing how it can be used, together with carefully chosen initial conditions, to simulate Post 2-tag systems, a known Turing complete problem.

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