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.

我们考虑了两个优化问题的逼近度，这两个优化问题称为最小开k-单子（Min-Open-k-Monopoly）和最小部分开k-单子（Min-P-Open-k-Monopoly），其中k是一个固定的正整数。在Min-Open-k-Monopoly中，目标是在给定图G =（V，E）中找到最小基数顶点集\（S \ subseteq V \），使得\（| N（v）\ cap S | \ GEQ \压裂{1} {2} | N（V）| + K \） ，对于每一个顶点v ∈ V。另一方面，给定一个图G =（V，E），在Min-P-Open-k-Monopoly中，需要找到最小基数顶点集\（S \ subseteq V \），使得\（| N（v）\ cap S | \ geq \ frac { 1} {2} | N（V）| + K \） ，对于每个v ∈ V ∖小号。我们证明Min-Open-k-Monopoly和Min-P-Open-k-Monopoly在\（O（\ log n）\）内是近似的。然后，我们证明这两个问题不能在\（（\ frac {1} {3}-\ epsilon）\ ln n \）和\（（\ frac {1} {4}-\ epsilon ）\ ln n \），对于任何𝜖 > 0，除非\（\ mathsf {NP} \ subseteq \ mathsf {Dtime}（n ^ {O（\ log \ log n）}）。\）对于4个正则图，我们证明这两个问题是A P X-完全的。Min-Open-1垄断可以在\（\ frac {26} {21} \ approx 1.2381 \）的范围内近似，而Min-P-Open-1-垄断可以在1.65153的因子范围内近似。对于ķ ≥2中，我们还存在近似算法这两个问题为（2 ķ + 2）规则的曲线图。