Upper and Lower Bounds on Approximating Weighted Mixed Domination

Abstract

A mixed dominating set of a graph $G=(V,E)$ is a mixed set $D$ of vertices and edges, such that for every edge or vertex, if it is not in $D$, then it is adjacent or incident to at least one vertex or edge in $D$. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is $NP$-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight $w_v$ and all edges are assigned the same nonnegative weight $w_e$, and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include: [1.] for $w_e\ge w_v$, a 2-approximation algorithm; [2.] for $w_e\ge 2w_v$, inapproximability within ratio 1.3606 unless $P=NP$ and within ratio 2 under UGC; [3.] for $2w_v>w_e\ge w_v$, inapproximability within ratio 1.1803 unless $P=NP$ and within ratio 1.5 under UGC; [4.] for $w_e<w_v$, inapproximability within ratio $(1-\epsilon )\ln |V|$ unless $P=NP$ for any $\epsilon >0$.

References in corpus (1)

• New parameterized algorithms for edge dominating set