An improved derandomized approximation algorithm for the max-controlled set problem

Fluminense Federal University, Institute of Computing, Rua Passo da Pátria 156, Bloco E, 24210-230, Niterói, RJ, Brazil; mart@dcc.ic.uff.br; fabio@ic.uff.br

Received: 16 June 2009
Accepted: 4 October 2010

Abstract

A vertex i of a graph G = (V,E) is said to be controlled by if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex is controlled by M. Given a set and two graphs G₁ = () and G₂ = () where , the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G = (V,E) (i.e., a graph where ) such that M is a monopoly in G = (V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G = (V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio + , where n=|V|>4. (The case is solved exactly by considering the parameterized version of the mcsp.) The algorithm is obtained through the use of randomized rounding and derandomization techniques based on the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.