On the approximability and exact algorithms for vector domination and related problems in graphs

TL;DR

This paper investigates the computational complexity of vector domination problems in graphs, proving inapproximability results, analyzing greedy strategies, and providing exact algorithms for specific graph classes.

Contribution

It establishes new inapproximability bounds, analyzes the effectiveness of greedy algorithms, and offers exact polynomial algorithms for certain graph classes.

Findings

Vector domination cannot be approximated within a factor of c ln n unless P=NP.

Greedy strategies achieve approximation factors of ln D + O(1).

Exact algorithms are provided for specific classes of graphs.

Abstract

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in S. In total vector domination, the requirement is extended to all vertices of the graph. We prove that these problems (and several variants thereof) cannot be approximated to within a factor of clnn, where c is a suitable constant and n is the number of the vertices, unless P = NP. We also show that two natural greedy strategies have approximation factors ln D+O(1), where D is the maximum degree of the input graph. We also provide exact polynomial time algorithms for several classes of graphs. Our results extend, improve, and unify several results previously known in the literature.