Minimum Consistent Subset in Interval Graphs and Circle Graphs

TL;DR

This paper investigates the Minimum Consistent Subset (MCS) problem on vertex-colored graphs, focusing on interval and circle graph classes. It introduces a leaf-bar-cover framework and a dynamic-programming approach to obtain a $O(n^9)$-time $(4α+2)$-approximation for MCS on interval graphs, constructing a near-optimal CS (ACS) from leaf-bar components. It also proves that MCS on circle graphs is APX-hard via a gap-preserving reduction from the dominating set problem on circle graphs. Collectively, these results delineate a concrete approximation bound for interval graphs while establishing hardness for circle graphs, guiding future exploration of PTAS and fixed-parameter tractable approaches. The methodology hinges on leaf bars, useful covers $Z_s$, and the ACS assembly, providing a constructive route from interval-graph structure to a provably close-to-optimal solution.

Abstract

In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c1, c2,..., c_α}$. We take a subset S of V, such that for every vertex v in V§, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such a S as a consistent subset. The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum size. It is established that MCS is NP-complete for general graphs, including planar graphs. We expand our study to interval graphs and circle graphs in an attempt to gain a complete understanding of the computational complexity of the \mcs problem across various graph classes. This work introduces an (4α+ 2)- approximation algorithm for MCS in interval graphs where αis the number of colors in the interval graphs. Later, we show that in circle graphs, MCS is APX-hard.

Figures (2)

• Figure 1: (A): The bar $s=(1..7)$ is a leaf bar. $l(s)=2$ and $r(s)=6$. $I[1]=I[3]=a$. (B): Boolean table M. $M[0,8]=1$, $M[0,9]=0$ because $(0..8)$ is a leaf bar but $(0..9)$ is not. (C): The 'green' color intervals in $Q_O$ are $a$ and $b$, and the greatest left endpoint is $x$. $I[x]=a$. The 'red' color interval in $Q_O$ is $d$. So, $Z_s=\{a,d\}$. (E): For 'red' color, $a$ and $b$ are the greatest left-endpoint interval and lowest right-endpoint interval in $Q_L$ and $Q_R$, respectively. Similarly, $c$ and $f$ are for 'green'. $Z_s=\{a,b,c,d\}$. (D): combining (C) and (E), we get $Z_s=\{c,f,g\}$. (F): $\{v_3, v_4, v_8\}$ is a CS and $\{v_7, v_8\}$ is an MCS. $\{v_1, v_8\}$ is also an MCS.
• Figure 2: An example reduction

Theorems & Definitions (22)

• Definition 1.1
• Definition 2.1
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4