Approximability of the upper chromatic number of hypergraphs

Csilla Bujtás, Z. Tuza

Research output: Contribution to journalArticle

Abstract

Abstract A C-coloring of a hypergraph H = (X, ε) is a vertex coloring φ : X → ℕ such that each edge E ∈ ε has at least two vertices with a common color. The related parameter χ¯(H), called the upper chromatic number of H, is the maximum number of colors in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E ∈ ε induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference n - χ¯(H), which is 2 + 2 ln(2m) on hypergraphs in general, and 1 + lnm on hypertrees. The latter ratio is essentially tight as we show that n - χ¯(H) cannot be approximated within (1 - ε)lnm on hypertrees (unless NP ⊆ DTIME(nO(loglogn))). Furthermore, χ¯(H) does not have O(n1 - ε)-approximation and cannot be approximated within additive error o(n) on the class of hypertrees (unless P = NP).

Original languageEnglish
Article number9927
Pages (from-to)1714-1721
Number of pages8
JournalDiscrete Mathematics
Volume338
Issue number10
DOIs
Publication statusPublished - May 31 2015

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Keywords

  • Approximation ratio
  • C-coloring
  • Hypergraph
  • Hypertree
  • Multiple hitting set
  • Upper chromatic number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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