Approximating the Minimum Independent Dominating Set in Perturbed Graphs

We investigate the minimum independent dominating set in perturbed graphs 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 n1− for any > 0. We prove that the size of the minimum independent dominating set in g(G, p), denoted as i(g(G, p)), is asymptotically almost surely in Θ(log |V |). Furthermore, we show that the probability of i(g(G, p)) ≥ √ 4|V | p is no more than 2 −|V |, and present a simple greedy algorithm of proven worst-case performance ratio √ 4|V | p and with polynomial expected running time.