Optimization Online


The (minimum) rank of typical fooling-set matrices

Mozhgan Pourmoradnasseri(mozhgan***at***ut.ee)
Dirk Oliver Theis(dotheis***at***ut.ee)

Abstract: A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\sqrt n$, for a matrix of order $n$. We ask for the typical minimum rank of a fooling-set matrix: For a fooling-set zero-nonzero pattern chosen at random, is the minimum rank of a matrix with that zero-nonzero pattern over a field $\FF$ closer to its lower bound $\sqrt{n}$ or to its upper bound $n$? We study random patterns with a given density $p$, and prove an $\Omega(n)$ bound for the cases when (a) $p$ tends to $0$ quickly enough; (b) $p$ tends to $0$ slowly, and $\abs{\FF}=O(1)$; (c) $p\in\lt]0,1\rt]$ is a constant. We leave open the case when $p\to 0$ slowly and $\FF$ is a large (e.g., $\FF=\GF(2^n)$, $\FF=\RR$).


Category 1: Combinatorial Optimization (Polyhedra )

Category 2: Combinatorial Optimization (Graphs and Matroids )


Download: [PDF]

Entry Submitted: 01/02/2017
Entry Accepted: 01/07/2017
Entry Last Modified: 01/02/2017

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society