On the Linear Extension Complexity of Regular n-gons

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope $P$. The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the $n$-gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound $2 \left\lceil \log_2(n) \right\rceil$ by one when $2^{k-1} < n \leq 2^{k-1}+2^{k-2}$ for some integer $k$. We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small $n$. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular $n$-gons (namely, $9 \leq n \leq 13$ and $21 \leq n \leq 24$) hence allowing for the first time to determine their extension complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the boolean rank of the slack matrices of n-gon

Low-Rank Boolean Matrix Approximation by Integer Programming

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best of our knowledge the first integer programming formulation that relies on only polynomially many variables and constraints, we discuss how to solve it computationally and report numerical tests on synthetic and real-world data

The Minrank of Random Graphs

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92). We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs $G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$, we show that $\mathsf{minrk}(G) = \Theta(n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$. This bound gives a near quadratic improvement over the previous best lower bound of $\Omega(\sqrt{n})$ (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs. Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits