The hardness of 3-uniform hypergraph coloring

Irit Dinur, Oded Regev, Clifford Smyth

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k > 2 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by [14], who also discussed the inherent difference between the k = 3 case and k ≥ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph [26]. We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

Original languageEnglish (US)
Title of host publicationAnnual Symposium on Foundations of Computer Science - Proceedings
EditorsD.C. Martin
Pages33-40
Number of pages8
StatePublished - 2002
EventThe 34rd Annual IEEE Symposium on Foundations of Computer Science - Vancouver, BC, Canada
Duration: Nov 16 2002Nov 19 2002

Other

OtherThe 34rd Annual IEEE Symposium on Foundations of Computer Science
CountryCanada
CityVancouver, BC
Period11/16/0211/19/02

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Coloring
Hardness
Color

ASJC Scopus subject areas

  • Hardware and Architecture

Cite this

Dinur, I., Regev, O., & Smyth, C. (2002). The hardness of 3-uniform hypergraph coloring. In D. C. Martin (Ed.), Annual Symposium on Foundations of Computer Science - Proceedings (pp. 33-40)

The hardness of 3-uniform hypergraph coloring. / Dinur, Irit; Regev, Oded; Smyth, Clifford.

Annual Symposium on Foundations of Computer Science - Proceedings. ed. / D.C. Martin. 2002. p. 33-40.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Dinur, I, Regev, O & Smyth, C 2002, The hardness of 3-uniform hypergraph coloring. in DC Martin (ed.), Annual Symposium on Foundations of Computer Science - Proceedings. pp. 33-40, The 34rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 11/16/02.
Dinur I, Regev O, Smyth C. The hardness of 3-uniform hypergraph coloring. In Martin DC, editor, Annual Symposium on Foundations of Computer Science - Proceedings. 2002. p. 33-40
Dinur, Irit ; Regev, Oded ; Smyth, Clifford. / The hardness of 3-uniform hypergraph coloring. Annual Symposium on Foundations of Computer Science - Proceedings. editor / D.C. Martin. 2002. pp. 33-40
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