Related papers: Minimizing the number of lattice points in a trans…
In this paper we give a new aggregation framework for linear Diophantine equations. In particular, we prove that an aggregated system of minimum size can be built in polynomial time. We also derive an analytic formula that gives the number…
We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b…
For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish…
In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be…
This paper presents algorithms for solving multiobjective integer programming problems. The algorithm uses Barvinok's rational functions of the polytope that defines the feasible region and provides as output the entire set of nondominated…
We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \,…
A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and…
For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant…
Let $A$ be an $(m \times n)$ integral matrix, and let $P=\{ x : A x \leq b\}$ be an $n$-dimensional polytope. The width of $P$ is defined as $ w(P)=min\{ x\in \mathbb{Z}^n\setminus\{0\} :\: max_{x \in P} x^\top u - min_{x \in P} x^\top v…
Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a…
In this paper, we present a deterministic algorithm for the closest vector problem for all l_p-norms, 1 < p < \infty, and all polyhedral norms, especially for the l_1-norm and the l_{\infty}-norm. We achieve our results by introducing a new…
Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…
Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimum positive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ... + e_k \sqrt{s_k} -t | $$ where $ s_1, s_2, ..., s_k$ are positive integers no larger than $n$, $t$ is an…
We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…
We consider a class of pattern matching problems where a normalising transformation is applied at every alignment. Normalised pattern matching plays a key role in fields as diverse as image processing and musical information processing…
In this paper, we address the problem of minimizing a convex function f over a convex set, with the extra constraint that some variables must be integer. This problem, even when f is a piecewise linear function, is NP-hard. We study an…
Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are…
We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which…
The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes…
We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m*) the number of lattice…