Approximating the domatic number

Uriel Feige, Magnús M. Halldórsson, Guy Kortsarz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices in a graph, δ the minimum degree, and Δ the maximum degree. Trivially, the domatic number is at most (δ + 1). We show that every graph has a domatic partition with (1 -o(1))(δ + 1)/ln n sets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1))ln n approximation algorithm for domatic number. We show that to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every ε > 0, a (1 - ε)ln n-approximation implies that NP ⊆ DTIME(nlog log n). This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also give a refined algorithm that gives a domatic partition of Ω(δ/ln Δ) sets. This implies an O(ln Δ) approximation for domatic number. We suspect that the true constant hidden by the Ω notation should be 1. As a step towards confirming this, we show that every graph of girth at least five and 5 large enough has domatic number at least (1 - o(1))δ/ln Δ.

Original languageEnglish
Title of host publicationProceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC 2000
Number of pages10
StatePublished - 2000
Event32nd Annual ACM Symposium on Theory of Computing, STOC 2000 - Portland, OR, United States
Duration: 21 May 200023 May 2000

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference32nd Annual ACM Symposium on Theory of Computing, STOC 2000
Country/TerritoryUnited States
CityPortland, OR


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