Minimum Consistent Subset in Trees and Interval Graphs

Document Type

Conference Article

Publication Title

Leibniz International Proceedings in Informatics Lipics

Abstract

In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph G, consisting of a vertex set V (G) of size n and an edge set E(G). Each vertex in V (G) is assigned a color from the set {1, 2, ⋯, c}. The objective is to determine a subset V′ ⊆ V (G) with minimum possible cardinality, such that for every vertex v ∈ V (G), at least one of its nearest neighbors in V′ (measured in terms of the hop distance) shares the same color as v. The decision problem, indicating whether there exists a subset V′ of cardinality at most l for some positive integer l, is known to be NP-complete even for planar graphs. In this paper, we establish that the MCS problem is NP-complete on trees. We also provide a fixed-parameter tractable (FPT) algorithm for MCS on trees parameterized by the number of colors (c) running in O(26cn6) time, significantly improving the currently best-known algorithm whose running time is O(24cn2c+3). In an effort to comprehensively understand the computational complexity of the MCS problem across different graph classes, we extend our investigation to interval graphs. We show that it remains NP-complete for interval graphs, thus enriching graph classes where MCS remains intractable.

DOI

10.4230/LIPIcs.FSTTCS.2024.7

Publication Date

12-5-2024

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