Minimum Selective Subset on Unit Disk Graphs and Circle Graphs

TL;DR

This work investigates the Minimum Selective Subset (MSS) problem on colored graphs, establishing a comprehensive hardness landscape and approximation results across graph classes. It proves log-APX-hardness for MSS on general graphs (even with two colors), NP-completeness for MSS on unit disk graphs, and APX-hardness for MSS on circle graphs, while delivering a PTAS for MSS on unit disk graphs that does not require explicit geometric input. A core contribution is a block-based PTAS framework that leverages 2-distance subsets and local/global solution merging, with MIS-based subroutines to bound local solutions. The study raises open questions on fixed-parameter tractability with respect to the color count or block count and invites further exploration of MSS in additional graph families such as circular-arc, chordal, and permutation graphs.

Abstract

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.

Figures (4)

• Figure 1: Colors: blue $\equiv$ cross, green $\equiv$ disk, red $\equiv$ square, and orange $\equiv$ circle. $V(G)=$$V_{blue}$$\cup$$V_{green}$$\cup$$V_{red}\cup V_{orange}$, where $V_{blue}=$$\{v_1\}$, $V_{green}$$=$$\{v_2$$,\dots,$$v_{9},v_{23},$$\dots,v_{26}\}$, $V_{red}=\{v_{10},\dots,v_{15},v_{27},\dots,v_{30}\}$, and $V_{orange}=\{v_{16},\dots,v_{22},v_{31},v_{32}\}$. $S=$$\{v_1,$$v_2,$$v_3,$$v_7,$$v_8,$$v_{9},v_{10},v_{11},v_{15},v_{16}\}$ is an MSS, and $S=\{v_1,$$v_4,$$v_5,$$v_7,v_8,v_{9},v_{12},v_{13},v_{15},v_{16}\}$ is also an MSS. Brown-dotted regions indicate the blocks. The complete list of blocks is $B_1=\{v_1\}$, $B_2=\{v_2,\dots,v_7,v_{23},\dots,v_{26}\}$, $B_3=\{v_{10},\dots,v_{15},v_{27},\dots,v_{30}\}$, $B_4=\{v_{16},\dots,v_{22},v_{31},v_{32}\}$, $B_{5}=\{v_{8}\}$, $B_{6}=\{v_{9}\}$. $B_{2,1}=\{v_2,v_3,v_6,v_7\}$, $B_{2,2}=\{v_4,v_5,v_{25},v_{26}\}$. $\{\{v_2\}, \{v_3\},\{v_7\}\}$ is a collection of 2-distance sets in $B_{2,1}$.
• Figure 2: Colors: blue $\equiv$ disk, red $\equiv$ square, and green $\equiv$ circle. (a) Reduction from an instance of Minimum Dominating Set problem to an instance of MSS problem when $c=2$. (b) Example of the reduction when $c=3$.
• Figure 3: Colors: blue $\equiv$ disk, red $\equiv$ circle, orange $\equiv$ fsquare, green $\equiv$ dash dotted circle, brown $\equiv$ dash dotted fsquare, darkred $\equiv$ square, darkgreen $\equiv$ cross, lightgreen $\equiv$ dotted circle, and violet $\equiv$ dotted square. An example of the reduction when $m=1$. Each blue disk in $U'$ is adjacent to a disk of a distinct color, different from all other colors in $U'$.
• Figure 4: Colors: blue $\equiv$ normal chord, and red $\equiv$ dotted chord. An example of the reduction: each chord of $G$ is colored blue in $H$. Each red chord in $H$ is adjacent only to a chord of blue color.

Theorems & Definitions (47)

• definition thmcounterdefinition: Selective Subset
• definition thmcounterdefinition: Block
• lemma thmcounterlemma
• lemma thmcounterlemma
• theorem thmcountertheorem
• remark thmcounterremark
• lemma thmcounterlemma
• theorem thmcountertheorem
• lemma thmcounterlemma
• remark thmcounterremark