TY - GEN

T1 - Approximating the domatic number

AU - Feige, Uriel

AU - Halldórsson, Magnús M.

AU - Kortsarz, Guy

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033701736&partnerID=8YFLogxK

U2 - 10.1145/335305.335321

DO - 10.1145/335305.335321

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AN - SCOPUS:0033701736

SN - 1581131844

SN - 9781581131840

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 134

EP - 143

BT - Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC 2000

Y2 - 21 May 2000 through 23 May 2000

ER -