Primal-dual based distributed algorithms for vertex cover with semi-hard capacities
Fabrizio Grandoni, Jochen Könemann, Alessandro Panconesi, Mauro Sozio · 2005
In this paper we consider the weighted, capacitated vertex cover problem with hard capacities (capVC). Here, we are given an undirected graph G = (V, E), non-negative vertex weightswtv for all vertices v ∈ V, and node-capacities Bv ≥ 1 for all v ∈ V. A feasible solution to a givencapVC instance consists of a vertex cover C ⊆ V. Each edge e ∈ E is assigned to one of its endpoints in C and the number of edges assigned to any vertex v ∈ C is at most Bv. The goal is to minimize the total weight of C. For a parameter ɛ> 0 we give a deterministic, distributed algorithm for thecapVC problem that computes a vertex cover C of weight at most (2+ɛ)·opt whereopt is the weight of a minimumweight feasible solution to the given instance. The number of edges assigned to any node v ∈ C is at most (4 + ɛ) · Bv. The running time of our algorithm is O(log(nW)/ɛ), where n is the number of nodes in the network and W = wtmax/wtmin is the ratio of largest to smallest weight. This result is complemented by a lower-bound saying that any distributed algorithm forcapVC which requires a poly-logarithmic number of rounds is bound to violate the capacity constraints by a factor two. The main feature of the algorithm is that it is derived in a systematic fashion starting from a primal-dual sequential algorithm.