0/1 Polytopes with quadratic Chvátal rank

Thomas Rothvoß, Laura Sanitá

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

11 Citations (Scopus)

Abstract

For a polytope P, the Chvátal closure P′ ⊆ P is obtained by simultaneously strengthening all feasible inequalities cx ≤ β (with integral c) to cz ≤ ⌊β⌋. The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvátal rank. If P ⊆ [0,1]n , then it is known that O(n2 log n) iterations always suffice (Eisenbrand and Schulz (1999)) and at least (1 + 1/e - o(1))n iterations are sometimes needed (Pokutta and Stauffer (2011)), leaving a huge gap between lower and upper bounds. We prove that there is a polytope contained in the 0/1 cube that has Chvátal rank Ω(n2), closing the gap up to a logarithmic factor. In fact, even a superlinear lower bound was mentioned as an open problem by several authors. Our choice of P is the convex hull of a semi-random Knapsack polytope and a single fractional vertex. The main technical ingredient is linking the Chvátal rank to simultaneous Diophantine approximations w.r.t. the ∥ · ∥1-norm of the normal vector defining P.

Original languageEnglish
Title of host publicationInteger Programming and Combinatorial Optimization - 16th International Conference, IPCO 2013, Proceedings
Pages349-361
Number of pages13
DOIs
Publication statusPublished - 2013
Externally publishedYes
Event16th Conference on Integer Programming and Combinatorial Optimization, IPCO 2013 - Valparaiso, Chile
Duration: 18 Mar 201320 Mar 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7801 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference16th Conference on Integer Programming and Combinatorial Optimization, IPCO 2013
CountryChile
CityValparaiso
Period18/03/1320/03/13

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