Approximating the Minimum Independent Dominating Set in Perturbed Graphs

Abstract

We investigate the minimum independent dominating set in perturbed graphs ${\mathfrak g}(G, p)$ of input graph G = (V, E), obtained by negating the existence of edges independently with a probability p > 0. The minimum independent dominating set (MIDS) problem does not admit a polynomial running time approximation algorithm with worst-case performance ratio of n ^1 − ε for any ε > 0. We prove that the size of the minimum independent dominating set in ${\mathfrak g}(G, p)$, denoted as $i({\mathfrak g}(G, p))$, is asymptotically almost surely in Θ(log|V|). Furthermore, we show that the probability of $i({\mathfrak g}(G, p)) \ge \sqrt{\frac{4|V|}{p}}$ is no more than 2^− |V|, and present a simple greedy algorithm of proven worst-case performance ratio $\sqrt{\frac{4|V|}{p}}$ and with polynomial expected running time.