相关论文: Basis Reduction, and the Complexity of Branch-and-…

The prototypical high-dimensional statistics problem entails finding a structured signal in noise. Many of these problems exhibit an intriguing phenomenon: the amount of data needed by all known computationally efficient algorithms far…

The Restricted Boltzmann Machine (RBM) is a stochastic neural network capable of solving a variety of difficult tasks such as NP-Hard combinatorial optimization problems and integer factorization. The RBM architecture is also very compact;…

We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that…

We describe and motivate a proposed new approach to lowerbounding the circuit complexity of boolean functions, based on a new formalization of "patterns" as elements of a special basis of the vector space of all truth table properties. We…

An influential 1990 paper of Hochbaum and Shanthikumar made it common wisdom that "convex separable optimization is not much harder than linear optimization" [JACM 1990]. We exhibit two fundamental classes of mixed integer (linear) programs…

There has recently been much progress on exact algorithms for the (un)weighted graph (bi)partitioning problem using branch-and-bound and related methods. In this note we present and improve an easily computable, purely combinatorial lower…

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \in Z^n with b' <= AUy <= b, y \in Z^n, where U is a unimodular matrix computed via basis reduction, to…

The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet well-understood,…

We propose a robust and scalable procedure for general optimization and inference problems on manifolds leveraging the classical idea of `median-of-means' estimation. This is motivated by ubiquitous examples and applications in modern data…

Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving…

Cutting-plane methods have enabled remarkable successes in integer programming over the last few decades. State-of-the-art solvers integrate a myriad of cutting-plane techniques to speed up the underlying tree-search algorithm used to find…

Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly…

For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special…

We present an efficient reduction from the Bounded integer programming (BIP) to the Subspace avoiding problem (SAP) in lattice theory. The reduction has some special properties with some interesting consequences. The first is the new upper…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is…

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The…

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…

Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such…