Minimum Consistent Subset in Trees and Interval Graphs

2024-04-23

Foundations of Software Technology and Theoretical Computer Science

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.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Social Networks and Social Media: Social mining and social search on the Web

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

• 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

• FPT algorithm for trees
• NP-completeness for interval graphs
• Improved time complexity bounds