Upper and Lower Bounds on Approximating Weighted Mixed Domination

TL;DR

This paper investigates approximation algorithms and hardness results for the weighted mixed dominating set problem in graphs, providing bounds and inapproximability ratios for various weight scenarios.

Contribution

It introduces new approximation algorithms and hardness bounds for the weighted mixed domination problem, extending understanding beyond the simple 2-approximation.

Findings

2-approximation algorithm for $w_e \,\geq\, w_v$

Hardness of approximation within ratio 1.3606 unless P=NP for $w_e \,\geq\, 2w_v$

Inapproximability within ratio 1.1803 unless P=NP for $2w_v > w_e \,\geq\, w_v$

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 $w_e\geq w_v$, a 2-approximation algorithm; [2.] for $w_e\geq 2w_v$, inapproximability within ratio 1.3606 unless $P=NP$ and within ratio 2 under UGC; [3.] for $2w_v > w_e\geq w_v$, 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 $\epsilon >0$.