Approximating the domatic number

Abstract

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.

A dependable Slepian-Wolf coding based clustering algorithm for data aggregation in wireless sensor networks

“…This problem is similar to the domatic partition problem in graph theory [8] , where a domatic partition is a partition of vertices in which each part is a dominating set. With the domatic partition, each vertex in the graph is either in the dominating set or has a neighbor in the set.…”

Optimal network locality in distributed virtualized data-centers

“…However, exact solutions can be computed for several classes of graphs [42]. We focus in this paper on one of these classes, namely the interval graph, because a linear-time algorithm is known for computing a domatic partition on any interval graph, and an interval graph is domatically full, i.e., k domatic partitions can be found in an interval graph with a minimal degree equal to k − 1.…”

“…These are works related to our algorithm development, although our objective is different. In these references, a given graph is partitioned into a number of partitions equal to the minimal graph degree, and the solution approaches are based on variants of a well-known randomized approximation algorithm [42]. In our case, we partition a complete weighted graph into k partitions in respect of the weights.…”

Sleeping Coordination for Comprehensive Sensing Using Isotonic Regression and Domatic Partitions

“…We believe that even networks with a few hundreds of nodes could be solved on a standard server. Even though attractive approximation algorithms are available [24], we decided to develop an ILP formulation to retain optimality so as to be able to assess the maximum extent of the gain of our approach. We solve our ILP using the commercial CPLEX package.…”