Minimum Consistent Subset in Trees and Interval Graphs
🤖 AI Summary
This paper studies the Minimum Consistent Subset (MCS) problem: given a graph with colored vertices, find a minimum vertex subset (V') such that every vertex has at least one same-color nearest neighbor under shortest-path distance (with a strict variant requiring *all* nearest neighbors to share its color). We first prove that MCS is NP-complete on trees parameterized by the number of colors (c). We then design a fixed-parameter tractable (FPT) algorithm based on tree decomposition and dynamic programming, running in (O(2^{6c} n^6)) time—significantly improving upon the previous best bound. Furthermore, we establish NP-completeness of MCS on interval graphs. Through structured reductions, a characterization of interval graphs, and refined parameterized techniques, we obtain tight complexity characterizations for MCS on both trees and interval graphs. Notably, our FPT algorithm features an exponential dependence solely on (c) and a polynomial factor of degree six, achieving both theoretical depth and practical scalability improvements.
Application Category
📝 Abstract
In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph $G=(V,E)$, consisting of a vertex set $V$ of size $n$ and an edge set $E$. Each vertex in $V$ is assigned a color from the set ${1,2,ldots, c}$. The objective is to determine a subset $V' subseteq V$ with minimum possible cardinality, such that for every vertex $v in V$, 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 for trees, when the number of colors $c$ is considered an input parameter, is NP-complete. We propose a fixed-parameter tractable (FPT) algorithm for MCS on trees running in $O(2^{6c}n^6)$ time, significantly improving the currently best-known algorithm whose running time is $O(2^{4c}n^{2c+3})$. 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.
Problem
Research questions and friction points this paper is trying to address.
NP-completeness of Minimum Consistent Subset in trees
FPT algorithm for MCS on trees with improved complexity
NP-completeness extension to interval graphs for MCS
Innovation
Methods, ideas, or system contributions that make the work stand out.
FPT algorithm for trees
NP-completeness for interval graphs
Improved time complexity bounds
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