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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, 6 the minimum degree, and A the maximum degree. Trivially, the domatic number is at most (6 + 1). We show that every graph has a domatic partition with (1 -o(1))(6 + 1)/lnn sets, and moreover, that such a domatic partition can be found in polynomiM time. This implies a (1 + o(1))lnn 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 e > 0, a (1-e)lnn-approximation implies that NP C DTIME(nl°gl°g'~). 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 ~2(5/In A) sets. This implies an O(ln A) approximation for domatic number. We suspect that the true constant hidden by the f/ 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))6/in Lx.

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We present a 1.5-approximation algorithm for the following NP-hard problem: given a connected graph G = (V, E) and an edge set E on V disjoint to E, find a minimum size subset of edges F ⊆ E such that (V, E ∪ F ) is 2-edge-connected. Our result improves and significantly simplifies the approximation algorithm with ratio 1.875 + ε of Nagamochi.

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