arXiv subjects

Joseph Paat

Publications and source records attributed to Joseph Paat.

At least 19 recordsLinked to original sources

arXiv2026

A characterization of maximal inhomogeneous-quadratic-free sets

The intersection cut framework is a versatile tool for generating valid inequalities in optimization. Its main ingredients are so-called $S$-free sets: convex sets whose interiors do not intersect a given set $S$. Among these, inclusion-wise maximal $S$-free sets are particularly important, as they yield the strongest intersection cuts. In the integer programming setting, maximal lattice-free sets are well studied and admit explicit characterizations. In the quadratic optimization context, Mu\~noz, Paat, and Serrano (2025) characterized maximal $S$-free sets when $S$ is defined by a homogeneous quadratic inequality. In this work, we characterize maximal $S$-free sets when $S$ is defined by an inhomogeneous quadratic inequality. As in the homogeneous case, our characterization is built using non-expansive functions. Together with the results in the homogeneous case, our results complete a characterization of $every$ maximal quadratic-free set via non-expansive functions.

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arXiv2025

The column number for 3-modular matrices

An integer-valued matrix $\mathbf{A}$ is $Δ$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $Δ$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $Δ$-modular matrix. Exact values for the column number are only known for $r \le 2$ or $Δ\le 2$. We prove that if $r$ is sufficiently large, then the maximum number of pairwise non-parallel columns of a rank-$r$, $3$-modular matrix is $\binom{r+1}{2} + 2(r-1)$. This settles a conjecture by Lee, Paat, Stallknecht, and Xu on the column number in the case $Δ= 3$. We complement this main result by showing that there are at least three $3$-modular matrices with pairwise non-isomorphic vector matroids that attain this upper bound. More generally, we show that if $r > Δ$, then the number of $Δ$-modular matrices with $\binom{r+1}{2} + (Δ-1)(r-1)$ pairwise non-parallel columns and pairwise non-isomorphic vector matroids is at least exponential in $\sqrtΔ$; previously only one matrix was known due to Lee et al.

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arXiv2024

Extended formulations for the integer hull of strictly $Δ$-modular cographic polyhedral cones

Conforti et al. give a compact extended formulation for a class of bimodular-constrained integer programs, namely those that model the stable set polytope of a graph with no disjoint odd cycles. We extend their techniques to design compact extended formulations for the integer hull of translated polyhedral cones whose constraint matrix is strictly $Δ$-modular and has rows that represent a cographic matroid. Our work generalizes the important special case from Conforti et al. concerning $4$-connected graphs with odd cycle transversal number at least $4$. We also discuss the necessity of our assumptions.

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arXiv2024

A characterization of unimodular hypergraphs with disjoint hyperedges

The incidence matrix of a graph is totally unimodular if and only if the graph is bipartite, i.e., it contains no odd cycles. We extend the characterization of total unimodularity to hypergraphs whose hyperedges of size at least four are pairwise disjoint, which we call disjoint hypergraphs. Disjoint hypergraphs have been used to model problems with fairness constraints that ensure balanced representation. We prove that total unimodularity for disjoint hypergraphs is equivalent to forbidding both odd cycles and structures that we call odd tree houses. Our result extends to disjoint mixed hypergraphs, whose incidence matrices have $\{0, \pm1\}$-entries. As a corollary, we resolve a special case of a conjecture on almost totally unimodular matrices, originally posed by Padberg and later modified by Cornu\'ejols and Zuluaga.

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arXiv2023

The MIP Workshop 2023 Computational Competition on Reoptimization

This paper describes the computational challenge developed for a computational competition held in 2023 for the $20^{\textrm{th}}$ anniversary of the Mixed Integer Programming Workshop. The topic of this competition was reoptimization, also known as warm starting, of mixed integer linear optimization problems after slight changes to the input data for a common formulation. The challenge was to accelerate the proof of optimality of the modified instances by leveraging the information from the solving processes of previously solved instances, all while creating high-quality primal solutions. Specifically, we discuss the competition's format, the creation of public and hidden datasets, and the evaluation criteria. Our goal is to establish a methodology for the generation of benchmark instances and an evaluation framework, along with benchmark datasets, to foster future research on reoptimization of mixed integer linear optimization problems.

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arXiv2023

A characterization of maximal homogeneous-quadratic-free sets

The intersection cut framework was introduced by Balas in 1971 as a method for generating cutting planes in integer optimization. In this framework, one uses a full-dimensional convex $S$-free set, where $S$ is the feasible region of the integer program, to derive a cut separating $S$ from a non-integral vertex of a linear relaxation of $S$. Among all $S$-free sets, it is the inclusion-wise maximal ones that yield the strongest cuts. Recently, this framework has been extended beyond the integer case in order to obtain cutting planes in non-linear settings. In this work, we consider the specific setting when $S$ is defined by a homogeneous quadratic inequality. In this 'quadratic-free' setting, every function $Γ: D^m \to D^n$, where $D^k$ is the unit disk in $\mathbb{R}^k$, generates a representation of a quadratic-free set. While not every $Γ$ generates a maximal quadratic free set, it is the case that every full-dimensional maximal quadratic free set is generated by some $Γ$. Our main result shows that the corresponding quadratic-free set is full-dimensional and maximal if and only if $Γ$ is non-expansive and satisfies a technical condition. This result yields a broader class of maximal $S$-free sets than previously known. Our result stems from a new characterization of maximal $S$-free sets (for general $S$ beyond the quadratic setting) based on sequences that 'expose' inequalities defining the $S$-free set.

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arXiv2023

Compressing Branch-and-Bound Trees

A branch-and-bound (BB) tree certifies a dual bound on the value of an integer program. In this work, we introduce the tree compression problem (TCP): Given a BB tree T that certifies a dual bound, can we obtain a smaller tree with the same (or stronger) bound by either (1) applying a different disjunction at some node in T or (2) removing leaves from T? We believe such post-hoc analysis of BB trees may assist in identifying helpful general disjunctions in BB algorithms. We initiate our study by considering computational complexity and limitations of TCP. We then conduct experiments to evaluate the compressibility of realistic branch-and-bound trees generated by commonly-used branching strategies, using both an exact and a heuristic compression algorithm.

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arXiv2022

On the column number and forbidden submatrices for $Δ$-modular matrices

An integer matrix $\mathbf{A}$ is $Δ$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $Δ$. The study of $Δ$-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by $Δ$-modular constraint matrices can be solved in polynomial time if $Δ$ is considered constant. The conjecture is only known to hold true when $Δ\in \{1,2\}$. In light of this conjecture, a natural question is to understand structural properties of $Δ$-modular matrices. We consider the column number question -- how many nonzero, pairwise non-parallel columns can a rank-$r$ $Δ$-modular matrix have? We prove that for each positive integer $Δ$ and sufficiently large integer $r$, every rank-$r$ $Δ$-modular matrix has at most $\binom{r+1}{2} + 80Δ^7 \cdot r$ nonzero, pairwise non-parallel columns, which is tight up to the term $80Δ^7$. This is the first upper bound of the form $\binom{r+1}{2} + f(Δ)\cdot r$ with $f$ a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a $Δ$-modular matrix. We believe this partial list may be of independent interest in future studies of $Δ$-modular matrices.

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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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arXiv2022

A Colorful Steinitz Lemma with Applications to Block Integer Programs

The Steinitz constant in dimension $d$ is the smallest value $c(d)$ such that for any norm on $\mathbb{R}^{ d}$ and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by $c(d)$. Grinberg and Sevastyanov prove that $c(d) \le d$ and that the bound of $d$ is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences. Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.

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arXiv2022

Polynomial upper bounds on the number of differing columns of $Δ$-modular integer programs

We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IP) with bounded determinants. For example, Artmann et al. showed that an IP can be solved in strongly polynomial time if the constraint matrix is bimodular, that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the $\ell_1$-distance between IP solutions and solutions of its linear relaxation. One of the first works to quantify the complexity of IPs with bounded determinants was that of Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller's bound to general determinants has been super-polynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first $\ell_1$-distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest.

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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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arXiv2021

The Integrality Number of an Integer Program

We introduce the integrality number of an integer program (IP) in inequality form. Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor $Δ$ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers $τ(Δ)$ and $κ(Δ)$ such that an IP with $n\geq τ(Δ)$ many variables and $n + κ(Δ)\cdot \sqrt{n}$ many inequality constraints can be solved via a MIP relaxation with fewer than $n$ integer constraints. From our results it follows that IPs defined by only $n$ constraints can be solved via a MIP relaxation with $O(\sqrtΔ)$ many integer constraints.

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arXiv2020

The distributions of functions related to parametric integer optimization

We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distribution of general functions related to integer optimization. There has been a significant amount of research focused around the extreme values that these functions can attain, however less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.

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arXiv2020

Constructing lattice-free gradient polyhedra in dimension two

Lattice-free gradient polyhedra can be used to certify optimality for mixed-integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. A classic result of Bell, Doignon, and Scarf states that a lattice-free gradient polyhedron with at most four facets exists in this setting. We present an algorithm for creating a sequence of gradient polyhedra, each of which has at most four facets, that finitely converges to a lattice-free gradient polyhedron. Each update requires constantly many gradient evaluations. Our updates imitate the gradient descent algorithm, and consequently, it yields a gradient descent type of algorithm for problems with two integer variables.

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arXiv2020

Improving proximity bounds using sparsity

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

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arXiv2019

Sparsity of integer solutions in the average case

We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Caratheodory number provided that the determinants in the data are small.

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arXiv2019

Non-unique lifting of integer variables in minimal inequalities

We explore the lifting question in the context of cut-generating functions. Most of the prior literature on this question focuses on cut-generating functions that have the unique lifting property. We develop a general theory for understanding the lifting question for cut-generating functions that do not necessarily have the unique lifting property.

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