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.
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Tong, W., Goebel, R., Lin, G. (2013). Approximating the Minimum Independent Dominating Set in Perturbed Graphs. In: Du, DZ., Zhang, G. (eds) Computing and Combinatorics. COCOON 2013. Lecture Notes in Computer Science, vol 7936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38768-5_24
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DOI: https://doi.org/10.1007/978-3-642-38768-5_24
Publisher Name: Springer, Berlin, Heidelberg
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