arXiv subjects

Yuri Faenza

Publications and source records attributed to Yuri Faenza.

41 records · Page 3Linked to original sources

arXiv2014

On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a $\bar c\,^{d}$-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix.

cs.CG↗

arXiv2013

Extended formulations, non-negative factorizations and randomized communication protocols

An extended formulation of a polyhedron $P$ is a linear description of a polyhedron $Q$ together with a linear map $π$ such that $π(Q)=P$. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis' factorization theorem [M. Yannakakis. Expressing combinatorial optimization problems by linear programs. {\em J. Comput. System Sci.}, 43(3):441--466 (1991)] provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $P$ equals the nonnegative rank of its slack matrix $S$. Moreover, Yannakakis also shows that the nonnegative rank of $S$ is at most $2^c$, where $c$ is the complexity of any \emph{deterministic} protocol computing $S$. In this paper, we show that the latter result can be strengthened when we allow protocols to be \emph{randomized}. In particular, we prove that the base-2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis' factorization theorem, this implies that the base-2 logarithm of the smallest size of an extended formulation of a polytope $P$ equals the minimum complexity of a randomized communication protocol computing the slack matrix of $P$ in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.

cs.DM↗

arXiv2012

On the convergence of the affine hull of the Chvátal-Gomory closures

Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chvátal-Gomory closure operator have to be performed on Q to obtain a polyhedron contained in the affine hull of P. We show that if P contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on n. On the other hand, we prove that if P is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.

math.OC↗

arXiv2011

A fast algorithm to remove proper and homogenous pairs of cliques (while preserving some graph invariants)

We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at most |E(G)| steps builds a new graph G' without proper and homogeneous pairs of cliques, and such that G and G' agree on the value of some relevant invariant (or property).

math.OC↗

arXiv2008

Extended Formulations for Packing and Partitioning Orbitopes

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one 1-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact that basically shifted-column inequalities suffice in order to describe those orbitopes linearly.

math.CO↗