An Improved Derandomized Approximation Algorithm for the Max-Controlled Set Problem

A vertex i of a graph G=(V,E) is said to be controlled by M ⊆ V 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 i ∈ V is controlled by M. Given a set M ⊆ V and two graphs G₁=(V,E₁) and G₂=(V,E₂) where E₁ ⊆ E₂, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G=(V,E) (i.e., a graph where E₁ ⊆ E ⊆ E₂) 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 $\frac{1}{2}+ \frac{1+\sqrt{n}}{2n-2}$, where n=|V|>4. (The case n ≤ 4 is solved exactly by considering the parameterized version of the mcsp.) The algoritm is obtained through the use of randomized rounding and derandomization techniques, namely the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.