Primal-dual based distributed algorithms for vertex cover with semi-hard capacities

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 weights wt_v for all vertices v ∈ V, and node-capacities B_v ≥ 1 for all v ∈ V. A feasible solution to a given capVC 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 B_v. The goal is to minimize the total weight of C. For a parameter ε > 0 we give a deterministic, distributed algorithm for the capVC problem that computes a vertex cover C of weight at most (2+ε) · opt where opt is the weight of a minimum-weight feasible solution to the given instance. The number of edges assigned to any node v ∈ C is at most (4 + ε) · B _v. The running time of our algorithm is O(log(nW)/ε), where n is the number of nodes in the network and W = wt_max/wt_min is the ratio of largest to smallest weight. This result is complemented by a lower-bound saying that any distributed algorithm for capVC 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.