Related papers: Extended Formulations for Packing and Partitioning…
Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…
With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing…
We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch conjecture and prove that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by a…
We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums $1$ improving upon Heller's classical bound. The proof uses Seymour's decomposition theorem. Such matrices are closely related to…
We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long…
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic…
A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have…
We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by…
Tverberg's theorem states that any set of $t(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Moreover, generic collections of fewer points cannot be so…
We envision programmable matter as a system of nano-scale agents (called particles) with very limited computational capabilities that move and compute collectively to achieve a desired goal. We use the geometric amoebot model as our…
We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able to compute the extension complexities of all 0/1-polytopes up to dimension 4. We…
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…
Plane partitions in the totally symmetric self-complementary symmetry class (TSSCPP) are known to be equinumerous with n x n alternating sign matrices, but no explicit bijection is known. In this paper, we give a bijection from these plane…
We show that packing axis-aligned unit squares into a simple polygon $P$ is NP-hard, even when $P$ is an orthogonal and orthogonally convex polygon with half-integer coordinates. It has been known since the early 80s that packing unit…
We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial…
An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…
In this work we study the polytope associated with a 0/1 integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining…
We consider polyhedral approximations of strictly convex compacta in finite dimensional Euclidean spaces (such compacta are also uniformly convex). We obtain the best possible estimates for errors of considered approximations in the…