In a connected simple graph G = (V(G),E(G)), each vertex is assigned one of c colors, where V(G) can be written as a union of a total of c subsets V_{1},...,V_{c} and V_{i} denotes the set of vertices of color i. A subset S of V(G) is called a selective subset if, for every i, every vertex v in V_{i} has at least one nearest neighbor in $S \cup (V(G) \setminus V_{i})$ that also lies in V_{i}. The Minimum Selective Subset (MSS) problem asks for a selective subset of minimum size. We show that the MSS problem is log-APX-hard on general graphs, even when c=2. As a consequence, the problem does not admit a polynomial-time approximation scheme (PTAS) unless P = NP. On the positive side, we present a PTAS for unit disk graphs, which works without requiring a geometric representation and applies for arbitrary c. We further prove that MSS remains NP-complete in unit disk graphs for arbitrary c. In addition, we show that the MSS problem is log-APX-hard on circle graphs, even when c=2.

翻译：在连通简单图 G = (V(G),E(G)) 中，每个顶点被分配 c 种颜色之一，其中 V(G) 可表示为 c 个子集 V_{1},...,V_{c} 的并集，且 V_{i} 表示颜色为 i 的顶点集合。若子集 S ⊆ V(G) 满足：对每个 i 及每个顶点 v ∈ V_{i}，v 在 $S \cup (V(G) \setminus V_{i})$ 中至少存在一个同属 V_{i} 的最近邻顶点，则称 S 为选择性子集。最小选择性子集（MSS）问题要求找出规模最小的选择性子集。我们证明，即使在 c=2 的情况下，MSS 问题在一般图上是 log-APX 难的。由此可推知，除非 P = NP，否则该问题不存在多项式时间近似方案（PTAS）。在积极方面，我们针对单位圆盘图提出一种 PTAS，该方案无需几何表示且适用于任意 c 值。我们进一步证明，对于任意 c，MSS 在单位圆盘图上仍是 NP 完全的。此外，我们还表明，即使在 c=2 的情况下，MSS 问题在圆图上也是 log-APX 难的。