Related papers: Sparsity of integer solutions in the average case
Sparse cutting-planes are often the ones used in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-&-bound more efficiently. However, how well can we approximate the integer…
A linear equation $E$ is said to be sparse if there is $c>0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterizing the sparse equations, first raised by Ruzsa in the…
We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to…
Cutting planes are crucial for the performance of branch-and-cut algorithms for solving mixed-integer programming (MIP) problems, and linear row aggregation has been successfully applied to better leverage the potential of several major…
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…
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulated problems, in which the columns of the constraint matrix are…
In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with…
A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally…
We consider a stochastic variant of the packing-type integer linear programming problem, which contains random variables in the objective vector. We are allowed to reveal each entry of the objective vector by conducting a query, and the…
In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.
It is well known that sparse approximation problem is \textsf{NP}-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the \emph{coherence} $\mu$ of the…
We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…
Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such…
This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and…
Sparse linear regression (SLR) is a well-studied problem in statistics where one is given a design matrix $X\in\mathbb{R}^{m\times n}$ and a response vector $y=X\theta^*+w$ for a $k$-sparse vector $\theta^*$ (that is, $\|\theta^*\|_0\leq…
In dictionary learning, also known as sparse coding, the algorithm is given samples of the form $y = Ax$ where $x\in \mathbb{R}^m$ is an unknown random sparse vector and $A$ is an unknown dictionary matrix in $\mathbb{R}^{n\times m}$…
In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron,…
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…
This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of…