On the Dominating Set Problem in Random Graphs

TL;DR

This paper investigates the computational complexity of the dominating set problem in random graphs, showing efficient algorithms for certain cases and complexity bounds related to approximation and parameterization.

Contribution

It provides expected-time algorithms for minimum dominating sets in random graphs and establishes complexity bounds for parameterized versions based on edge probability.

Findings

Expected $2^{O(\log^{2} n)}$ time for minimum dominating set

Complexity bounds linked to approximation ratios

Efficient algorithms when edge probability varies with $n$

Abstract

In this paper, we study the {\sc Dominating Set} problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of $p$, where $p$ is a positive constant less than $1$. We show that, given a random graph in $n$ vertices, a minimum dominating set in the graph can be computed in expected $2^{O(\log_{2}^{2}{n})}$ time. For the parameterized dominating set problem, we show that it cannot be solved in expected $O(f(k)n^{c})$ time unless the minimum dominating set problem can be approximated within a ratio of $o(\log_{2}n)$ in expected polynomial time, where $f(k)$ is a function of the parameter $k$ and $c$ is a constant independent of $n$ and $k$. In addition, we show that the parameterized dominating set problem can be solved in expected $O(f(k)n^{c})$ time when the probability $p$ depends on $n$ and equals to $\frac{1}{g(n)}$, where $g(n)< n$ is a monotonously increasing function of $n$ and its value approaches infinity when $n$ approaches infinity.