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 n 1 - ? 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 ? ( 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.