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

ABSTRACTIn 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 wtv for all vertices v ∈ V, and node-capacities Bv ≥ 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 Bv. 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)• Bv. The running time of our algorithm is O(log (n W)/ε), where n is the number of nodes in the network and W=wtmax/weightmin 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.