arXiv subjects

Robert Weismantel

Publications and source records attributed to Robert Weismantel.

At least 19 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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arXiv2026

Alternating Linear Minimization: Revisiting von Neumann's alternating projections

In 1933 von Neumann proved a beautiful result that one can approximate a point in the intersection of two convex sets by alternating projections, i.e., successively projecting on one set and then the other. This algorithm assumes that one has access to projection operators for both sets. In this note, we consider the much weaker setup where we have only access to linear minimization oracles over the convex sets and present an algorithm to find a point in the intersection of two convex sets.

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arXiv2026

A Threshold Phenomenon for the Shortest Lattice Vector Problem in the Infinity Norm

One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with lattices of the form $A\mathbb{Z}^n$, where $A$ has integer entries and is of full column rank. Finding a shortest vector is NP-hard. We show that this task is fixed parameter tractable in the parameter $Δ$, the largest absolute value of the determinant of a full rank submatrix of $A$. The algorithm is based on a structural result that can be interpreted as a threshold phenomenon: whenever the dimension $n$ exceeds a certain value determined only by $Δ$, then a shortest lattice vector attains an infinity norm value of one. This threshold phenomenon has several applications. In particular, it reveals that integer optimal solutions lie on faces of the given polyhedron whose dimensions are bounded only in terms of $Δ$.

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arXiv2026

Sparse Approximation in Lattices and Semigroups

This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number $k < n$, to approximate $b$ with $Ay$ where $y$ is an integer or non-negative integer solution with at most $k$ non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter $k$, then the paper explains why the quality of the approximation increases exponentially as $k$ goes to $n$. This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO).

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arXiv2024

Forall-exist statements in pseudopolynomial time

Given a convex set $Q \subseteq R^m$ and an integer matrix $W \in Z^{m \times n}$, we consider statements of the form $ \forall b \in Q \cap Z^m$ $\exists x \in Z^n$ s.t. $Wx \leq b$. Such statements can be verified in polynomial time with the algorithm of Kannan and its improvements if $n$ is fixed and $Q$ is a polyhedron. The running time of the best-known algorithms is doubly exponential in~$n$. In this paper, we provide a pseudopolynomial-time algorithm if $m$ is fixed. Its running time is $(m Δ)^{O(m^2)}$, where $Δ= \|W\|_\infty$. Furthermore it applies to general convex sets $Q$.

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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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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

Sparse Approximation Over the Cube

This paper presents an anlysis of the NP-hard minimization problem $\min \{\|b - Ax\|_2: \ x \in [0,1]^n, | \text{supp}(x) | \leq σ\}$, where $\text{supp}(x) = \{i \in [n]: x_i \neq 0\}$ and $σ$ is a positive integer. The object of investigation is a natural relaxation where we replace $| \text{supp}(x) | \leq σ$ by $\sum_i x_i \leq σ$. Our analysis includes a probabilistic view on when the relaxation is exact. We also consider the problem from a deterministic point of view and provide a bound on the distance between the images of optimal solutions of the original problem and its relaxation under $A$. This leads to an algorithm for generic matrices $A \in \mathbb{Z}^{m \times n}$ and achieves a polynomial running time provided that $m$ and $\|A\|_{\infty}$ are fixed.

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arXiv2022

Minimizing a low-dimensional convex function over a high-dimensional cube

For a matrix $W \in \mathbb{Z}^{m \times n}$, $m \leq n$, and a convex function $g: \mathbb{R}^m \rightarrow \mathbb{R}$, we are interested in minimizing $f(x) = g(Wx)$ over the set $\{0,1\}^n$. We will study separable convex functions and sharp convex functions $g$. Moreover, the matrix $W$ is unknown to us. Only the number of rows $m \leq n$ and $\|W\|_{\infty}$ is revealed. The composite function $f(x)$ is presented by a zeroth and first order oracle only. Our main result is a proximity theorem that ensures that an integral minimum and a continuous minimum for separable convex and sharp convex functions are always "close" by. This will be a key ingredient to develop an algorithm for detecting an integer minimum that achieves a running time of roughly $(m \| W \|_{\infty})^{\mathcal{O}(m^3)} \cdot \text{poly}(n)$. In the special case when $(i)$ $W$ is given explicitly and $(ii)$ $g$ is separable convex one can also adapt an algorithm of Hochbaum and Shanthikumar. The running time of this adapted algorithm matches with the running time of our general algorithm.

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arXiv2022

Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $Ω(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first non-trivial lower bound on the number of integer variables needed in such settings.

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arXiv2022

Minimum-Cost Flows Over Time

In this paper we show that every maximum minimum-cost flow over time problem has an optimal solution with a repeated structure if the given time horizon is large enough.

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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

On lattice width of lattice-free polyhedra and height of Hilbert bases

We study the lattice width of lattice-free polyhedra given by $\mathbf{A}\mathbf{x}\leq\mathbf{b}$ in terms of $Δ(\mathbf{A})$, the maximal $n\times n$ minor in absolute value of $\mathbf{A}\in\mathbb{Z}^{m\times n}$. Our main contribution is to link the lattice width of lattice-free polyhedra to the height of Hilbert bases and to the diameter of finite abelian groups. This leads to a bound on the lattice width of lattice-free pyramids which solely depends on $Δ(\mathbf{A})$ provided a conjecture regarding the height of Hilbert bases holds. Further, we exploit a combination of techniques to obtain novel bounds on the lattice width of simplices. A second part of the paper is devoted to a study of the above mentioned Hilbert basis conjecture. We give a complete characterization of the Hilbert basis if $Δ(\mathbf{A}) = 2$ which implies the conjecture in that case and prove its validity for simplicial cones.

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arXiv2021

Notes on $\{a,b,c\}$-Modular Matrices

Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$ subdeterminants of $A$ in absolute value is $\{a,b,c\}$. We will succeed in solving this problem in polynomial time unless $A$ possesses a duplicative relation, that is, $A$ has nonzero $n \times n$ subdeterminants $k_1$ and $k_2$ satisfying $2 \cdot |k_1| = |k_2|$. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over $\{a,b,c\}$-modular constraint matrices for any constants $a$, $b$ and $c$.

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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

Efficient sequential and parallel algorithms for multistage stochastic integer programming using proximity

We consider the problem of solving integer programs of the form $\min \{\,c^\intercal x\ \colon\ Ax=b, x\geq 0\}$, where $A$ is a multistage stochastic matrix in the following sense: the primal treedepth of $A$ is bounded by a parameter $d$, which means that the columns of $A$ can be organized into a rooted forest of depth at most $d$ so that columns not bound by the ancestor/descendant relation in the forest do not have non-zero entries in the same row. We give an algorithm that solves this problem in fixed-parameter time $f(d,\|A\|_{\infty})\cdot n\log^{O(2^d)} n$, where $f$ is a computable function and $n$ is the number of rows of $A$. The algorithm works in the strong model, where the running time only measures unit arithmetic operations on the input numbers and does not depend on their bitlength. This is the first fpt algorithm for multistage stochastic integer programming to achieve almost linear running time in the strong sense. For the case of two-stage stochastic integer programs, our algorithm works in time $2^{(2\|A\|_\infty)^{O(r(r+s))}}\cdot n\log^{O(rs)} n$. The algorithm can be also parallelized: we give an implementation in the PRAM model that achieves running time $f(d,\|A\|_{\infty})\cdot \log^{O(2^d)} n$ using $n$ processors. The main conceptual ingredient in our algorithms is a new proximity result for multistage stochastic integer programs. We prove that if we consider an integer program $P$, say with a constraint matrix $A$, then for every optimum solution to the linear relaxation of $P$ there exists an optimum (integral) solution to $P$ that lies, in the $\ell_{\infty}$-norm, within distance bounded by a function of $\|A\|_{\infty}$ and the primal treedepth of $A$. On the way to achieve this result, we prove a generalization and considerable improvement of a structural result of Klein for multistage stochastic integer programs.

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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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