Approximation Algorithms for Highly Connected Multi-dominating Sets in Unit Disk Graphs

Abstract

Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((k, m)-CDS) is defined as a subset S of V such that each node in V \ S has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (k, m)-CDS problem is to find a minimum weight (k, m)-CDS in a given node-weighted graph. The problem is called the unweighted (k, m)-CDS problem if the objective is to minimize the cardinality of a (k, m)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. In this paper, we consider the case in which k ≤ m, and we present a simple O(k5 k )-approximation algorithm for the unweighted (k, m)-CDS problem, and a primal-dual O(k 2 log k)-approximation algorithm for the weighted (k, m)-CDS problem.A CDS does not give a fault-tolerant virtual backbone network. This is because a CDS is only required to be connected, and each node outside a CDS is required to have only one neighbor in the CDS. Hence, if a backbone node fails, the virtual backbone network may be disconnected, or a nonbackbone node may lose access to the virtual backbone network. To overcome this disadvantage, Dai and Wu [10] proposed replacing a CDS by a k-connected k-dominating set, and they addressed the problem of finding a minimum k-connected k-dominating set in a unit disk graph. For a graph with the node set V , a subset S of V is called k-connected if the subgraph induced by S is kconnected (i.e., it is connected even if any k − 1 nodes are removed), and is called k-dominating if each node v ∈ V \ S has k neighbors in S. Triggered by their study, much attention has been paid to this problem, extending the notion of a k-connected k-dominating set to a more-general k-connected m-dominating set ((k, m)-CDS).The problem of finding a minimum cardinality (k, m)-CDS in a unit disk graph is called the unweighted (k, m)-CDS problem. If each node is given a nonnegative weight, and the objective is to minimize the weight of a (k, m)-CDS, then this is called the weighted (k, m)-CDS problem. As for the unweighted (k, m)-CDS problem, several constant-approximation algorithms were given for k ≤ 3 [21,22,26,27,30]. As for the weighted (k, m)-CDS problem, there are several constantapproximation algorithms for k = m = 1 [2, 31], but no approximation algorithm was known for the case of (k, m) = (1, 1) before our study (see Section 2 for more literature reviews).After these previous studies, a natural question arises as to whether there is a constantapproximation algorithm for the unweighted (k, m)-CDS problem with k ≥ 4, and for the weighted problem with (k, m) = (1, 1). For the unweighted problem, this question has been already addressed in both [26] and [27].

Improved approximation algorithms for k-connected m-dominating set problems

“…We consider the following problem for m ≥ k both in general graphs and in unit-disc graphs. For motivation we refer the reader to recent papers of Zhang, Zhou, Mo, and Du [10] and of Fukunaga [2], where they obtained in unit-disc graphs ratios O(k 3 ln k) and O(k 2 ln k), respectively. This was improved to O(k ln k) in [9], where is also given ratio O(k 2 ln n) in general graphs.…”

“…Similarly, a graph is k-T -connected if it has k internally node disjoint st-paths for every s, t ∈ T . A reason why the case m ≥ k is easier than the case m < k is given in the following statement (a proof can be found in [10,2,9]).…”

“…The above lemma implies that in the case m ≥ k (k, m)-CDS partial solutions have the property that the union of a partial solution and a feasible solution is always feasible -this enables to construct the solution iteratively. Specifically, most algorithms for the case m ≥ k start by computing just an m-dominating set T ; the best ratios for m-Dominating Set are ln(∆+m) in general graphs [1] and O(1) in unit disc graphs [2]. By invoking just these ratios, Lemma 1 enables to reduce (k, m)-CDS with m ≥ k to following (node weighted) problem:…”

“…This implies that the node weighted case is reduced with a loss of factor O(k) to the case of node induced edge costs -when c uv = w u +w v for every edge e = uv ∈ E. The edge costs version of Subset k-Connectivity admits ratio O(k 2 ln k), which gives ratio O(k 3 ln k) for (k, m)-CDS with m ≥ k in unit disc graphs. Fukunaga [2] obtained ratio O(k 2 ln k) using a different approachhe considered the Rooted Subset Connectivity Augmentation problem, when G[T ] is ℓ-(T, r)-connected and we seek a minimum weight S ⊆ V \ T such that G[T ∪ S] is (ℓ + 1)-(T, r)-connected. In [7] it is shown that the augmentation problem decomposes into O(k) "uncrossable" subproblems, and Fukunaga [2] designed a primal-dual O(1)-approximation algorithm for such an uncrossable subproblem in unit disc graphs.…”

“…Fukunaga [2] obtained ratio O(k 2 ln k) using a different approachhe considered the Rooted Subset Connectivity Augmentation problem, when G[T ] is ℓ-(T, r)-connected and we seek a minimum weight S ⊆ V \ T such that G[T ∪ S] is (ℓ + 1)-(T, r)-connected. In [7] it is shown that the augmentation problem decomposes into O(k) "uncrossable" subproblems, and Fukunaga [2] designed a primal-dual O(1)-approximation algorithm for such an uncrossable subproblem in unit disc graphs. This gives ratio O(ℓ) for Rooted Subset Connectivity Augmentation in unit disc graphs.…”