License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.42
URN: urn:nbn:de:0030-drops-94461
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9446/
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Haviv, Ishay

On Minrank and Forbidden Subgraphs

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Abstract

The minrank over a field F of a graph G on the vertex set {1,2,...,n} is the minimum possible rank of a matrix M in F^{n x n} such that M_{i,i} != 0 for every i, and M_{i,j}=0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H,F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this paper we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Omega(sqrt{n}/log n) for the triangle H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H,R) >= n^delta for some delta = delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudlák, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.

BibTeX - Entry

@InProceedings{haviv:LIPIcs:2018:9446,
  author =	{Ishay Haviv},
  title =	{{On Minrank and Forbidden Subgraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial  Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{42:1--42:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9446},
  URN =		{urn:nbn:de:0030-drops-94461},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.42},
  annote =	{Keywords: Minrank, Forbidden subgraphs, Shannon capacity, Circuit Complexity}
}

Keywords: Minrank, Forbidden subgraphs, Shannon capacity, Circuit Complexity
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)
Issue Date: 2018
Date of publication: 13.08.2018

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