Related papers: Minimizing the number of lattice points in a trans…
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…
Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point…
We introduce polyhedra circuits. Each polyhedra circuit characterizes a geometric region in $\mathbb{R}^d$. They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of…
We study integer linear programs (ILP) of the form $\min\{c^\top x\ \vert\ Ax=b,l\le x\le u,x\in\mathbb Z^n\}$ and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the…
We consider the Min-$r$-Lin$(Z_m)$ problem: given a system $S$ of length-$r$ linear equations modulo $m$, find $Z \subseteq S$ of minimum cardinality such that $S-Z$ is satisfiable. The problem is NP-hard and UGC-hard to approximate in…
Counting lattice points within a rational polytope is a foundational problem with applications across mathematics and computer science. A key approach is Barvinok's algorithm, which decomposes the lattice point generating function of cones…
The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in R^N, and their…
Given two point sets in the plane, we study the minimization of the bottleneck distance between a point set B and an equally-sized subset of a point set A under translations. We relate this problem to a Voronoi-type diagram and derive…
We extend the Barvinok-Woods algorithm for enumerating projections of integer points in polytopes to unbounded polyhedra. For this, we obtain a new structural result on projections of semilinear subsets of the integer lattice. We extend the…
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…
In this article, we investigate the multi-parametric matroid problem. The weights of the elements of the matroid's ground set depend linearly on an arbitrary but fixed number of parameters, each of which is taken from a real interval. The…
Let $X$ be a finite set in $Z^d$. We consider the problem of optimizing linear function $f(x) = c^T x$ on $X$, where $c\in Z^d$ is an input vector. We call it a problem $X$. A problem $X$ is related with linear program $\max\limits_{x \in…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in $\mathbb{R}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset…
We consider the nonlinear integer programming problem of minimizing a quadratic function over the integer points in variable dimension satisfying a system of linear inequalities. We show that when the Graver basis of the matrix defining the…
For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a…
The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces `short' simplex paths from any given vertex to an optimal one. We consider a lattice polytope $P$ contained in $[0,k]^n$ and…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
We introduce the parametric matroid one-interdiction problem. Given a matroid, each element of its ground set is associated with a weight that depends linearly on a real parameter from a given parameter interval. The goal is to find, for…
Nonconvex optimization problems with an L1-constraint are ubiquitous, and are found in many application domains including: optimal control of hybrid systems, machine learning and statistics, and operations research. This paper shows that…