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 <i>G=(V,E)</i>, non-negative vertex weights <i>wt<inf>v</inf></i> for all vertices <i>v ∈ V</i>, and node-capacities <i>B<inf>v</inf> ≥ 1</i> for all <i>v ∈ V</i>. A feasible solution to a given capVC instance consists of a vertex cover <i>C ⊆ V</i>. Each edge <i>e ∈ E</i> is assigned to one of its endpoints in <i>C</i> and the number of edges assigned to any vertex <i>v ∈ C</i> is at most <i>B<inf>v</inf></i>. The goal is to minimize the total weight of <i>C</i>.For a parameter <i>ε>0</i> we give a deterministic, distributed algorithm for the capVC problem that computes a vertex cover <i>C</i> of weight at most <i>(2+ε) • opt</i> where <i>opt</i> is the weight of a minimum-weight feasible solution to the given instance. The number of edges assigned to any node <i>v ∈ C</i> is at most <i>(4+ε)• B<inf>v</inf></i>. The running time of our algorithm is <i>O(log (n W)/ε)</i>, where n is the number of nodes in the network and <i>W=wt<inf>max</inf>/weight<inf>min</inf></i> 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.

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