Minimum Strict Consistent Subset in Paths, Spiders, Combs and Trees

TL;DR

This paper studies the Minimum Strict Consistent Subset (MSCS) problem on colored graphs, formalizing CS and SCS using colors $c_1,\dots,c_\alpha$ and a nearest-neighbor criterion. It proves MSCS is NP-hard on general graphs via a dominating-set reduction and surveys related results on MCS/MCSS. It then provides a 2-approximation for MSCS in trees and an $O(n^4)$ dynamic-programming algorithm for MSCS on trees, along with faster polynomial-time algorithms for MSCS in paths, spiders, and combs ($O(n)$, $O(n^2)$, and $O(n^3)$ respectively). Finally, it discusses open problems for planar graphs and broader classes, outlining directions for future work in algorithm design and complexity.

Abstract

Let G be a simple connected graph with vertex set V(G) and edge set E(G. Each vertex of V(G) is colored by a color from the set of colors {c_1, c_2,\dots, c_α}. We take a subset S of V(G), such that for every vertex v in V(G)§, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such an S as a consistent subset (CS). The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum cardinality. It is established that MCS is NP-complete for general graphs, including planar graphs. The strict consistent subset is a variant of consistent subset problems. We take a subset S^{\prime} of V(G), such that for every vertex v in V(G)§^{\prime}, all the vertices in its set of nearest neighbors in S^{\prime} have the same color as that of v. We refer to such an S^{\prime} as a strict consistent subset (SCS). The Minimum Strict Consistent Subset (MSCS) problem is the computation of a strict consistent subset of the minimum cardinality. We demonstrate that MSCS is NP-hard for general graphs using a reduction from dominating set problems. We construct a 2-approximation algorithm and a polynomial-time algorithm in trees. Lastly, we conclude the faster polynomial-time algorithms in paths, spiders, and combs.

Figures (8)

• Figure 1: (A) Blocks are $B_1, \dots B_6$. (B) The corresponding block tree.
• Figure 2: (A), (B) and (C) are the examples of MCS, MCSS and MSCS for a two-colored unweighted tree, respectively.
• Figure 3: (A) Illustration of Lemma \ref{['lemma1']}. (B) The reduced graph $G^{\prime}$.
• Figure 4: (A) The tree $\mathrm{T}\xspace$ , and (B) the tree $\mathrm{T}\xspace ^{\prime}$.
• Figure 5: Solving $\mathrm{T}\xspace (x, y)$ recursively in terms of $\mathrm{T}\xspace (x, v^{\prime})$ and $\mathrm{T}\xspace (z, v ^{\prime} )$ where $z$ is a valid pair for $x$.

Theorems & Definitions (12)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Lemma 5.1
• proof