Approximability of open k-monopoly problems

Abstract

We consider approximability of two optimization problems called Minimum Open k-Monopoly (Min-Open-k-Monopoly) and Minimum Partial Open k-Monopoly (Min-P-Open-k-Monopoly), where k is a fixed positive integer. The objective, in Min-Open-k-Monopoly, is to find a minimum cardinality vertex set $S \subseteq V$ in a given graph G = (V,E) such that $|N(v) \cap S| \geq \frac {1}{2} |N(v)| + k$, for every vertex v ∈ V. On the other hand, given a graph G = (V,E), in Min-P-Open-k-Monopoly it is required to find a minimum cardinality vertex set $S \subseteq V$ such that $|N(v) \cap S| \geq \frac {1}{2} |N(v)| + k$, for every v ∈ V ∖ S. We prove that Min-Open-k-Monopoly and Min-P-Open-k-Monopoly are approximable within a factor of $O(\log n)$. Then, we show that these two problems cannot be approximated within a factor of $(\frac {1}{3} - \epsilon )\ln n$ and $(\frac {1}{4} - \epsilon )\ln n$, respectively, for any 𝜖 > 0, unless $\mathsf {NP} \subseteq \mathsf {Dtime}(n^{O(\log \log n)}).$ For 4-regular graphs, we prove that these two problems are APX-complete. Min-Open-1-Monopoly can be approximated within a factor of $\frac {26}{21} \approx 1.2381$ where as Min-P-Open-1-Monopoly can be approximated within a factor of 1.65153. For k ≥ 2, we also present approximation algorithms for these two problems for (2k + 2)-regular graphs.