Almost separable matrices

Matthew Aldridge*, Leonardo Baldassini, Karen Gunderson

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

11 Citations (Scopus)
231 Downloads (Pure)

Abstract

An m x n matrix A with column supports {Si} is k-separable if the disjunctions UiεkSi are all distinct over all sets K of cardinality k. While a simple counting bound shows that m > klog2n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of ‘almost k-separability’, which requires that the disjunctions are ‘mostly distinct’. We show using a random construction that these matrices exist with m = O(log n) rows, which is optimal for k = O(n1-β). Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.

Original languageEnglish
Pages (from-to)215-236
Number of pages22
JournalJournal of Combinatorial Optimization
Volume33
Issue number1
Early online date12 Oct 2015
DOIs
Publication statusPublished - Jan 2017

Keywords

  • Cover-free families
  • Disjunct matrices
  • Group testing
  • Probabilistic method
  • Separable matrices
  • Union-free families

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