Approximation Algorithms for the b-Edge Dominating Set Problem and Its Related Problems

The edge dominating set problem is one of the fundamental covering problems in the field of combinatorial optimization. In this paper, we consider the b-edge dominating set problem, a generalized version of the edge dominating set problem. In this version, we are given a simple undirected graph G=(V,E) and a demand vector $b \in \mathbb{Z}_+^E$. A set F of edges in G is called a b-edge dominating set if each edge e ∈ E is adjacent to at least b(e) edges in F, where we allow F to contain multiple copies of edges in E. Given a cost vector $w \in \mathbb{Q}_+^E$, the problem asks to find a minimum cost of a b-edge dominating set. We first show that there is a $\frac{8}{3}$-approximation algorithm for this problem. We then consider approximation algorithms for other related problems.