Minimum Consistent Subset Problem is NP-Complete for Trees and Interval Graphs

Core Concepts

The Minimum Consistent Subset (MCS) problem is NP-complete for trees and interval graphs, even when the number of colors is considered as a parameter. However, the problem is fixed-parameter tractable for trees.

Abstract

The key insights and findings of the content are:

The Minimum Consistent Subset (MCS) problem is shown to be NP-complete for trees, even when the number of colors is considered as a parameter. This is a significant result, as there are few naturally occurring problems known to be NP-hard on trees.

The authors present a fixed-parameter tractable (FPT) algorithm for solving the MCS problem on trees, which runs in O(26^c * n^6) time, significantly improving the previously best-known algorithm with a running time of O(24^c * n^(2c+3)).

The authors also show that the MCS problem remains NP-complete for interval graphs, further expanding the set of graph classes where the problem is intractable.

The hardness results for both trees and interval graphs demonstrate the computational complexity of the MCS problem across different graph classes, providing a comprehensive understanding of its tractability.

The FPT algorithm for trees highlights the possibility of efficient solutions for the MCS problem on certain restricted graph classes, despite its general intractability.

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Key Insights Distilled From

Minimum Consistent Subset in Trees and Interval Graphs

by Aritra Banik... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15487.pdf

Deeper Inquiries

Can the fixed-parameter tractable algorithm for the MCS problem on trees be further optimized or generalized to other graph classes

The fixed-parameter tractable algorithm for the MCS problem on trees can potentially be optimized by exploring different parameterizations or by refining the dynamic programming approach. One possible optimization could involve incorporating heuristics or pruning techniques to reduce the search space and improve the efficiency of the algorithm. Additionally, the algorithm could be generalized to other graph classes by adapting the dynamic programming framework to suit the specific characteristics of those graphs. For instance, for graph classes with certain structural properties, modifications to the algorithm may be necessary to account for different connectivity patterns or coloring constraints.

Are there any other graph classes, besides trees and interval graphs, where the MCS problem remains computationally tractable

While the MCS problem has been shown to be NP-complete for trees and interval graphs, there are other graph classes where the problem remains computationally tractable. For example, the MCS problem may be more manageable for specific subclasses of graphs such as paths, cycles, or bipartite graphs due to their inherent structural properties. In these cases, the problem may exhibit polynomial-time complexity or have efficient algorithms that can find optimal solutions without exponential time complexity. By identifying and studying graph classes where the MCS problem is tractable, researchers can gain insights into the interplay between graph structures and computational complexity.

What are the potential applications of the MCS problem, and how can the insights from this work be leveraged to improve real-world decision-making tasks

The Minimum Consistent Subset (MCS) problem has various potential applications in real-world scenarios, particularly in the field of pattern recognition, data analysis, and decision-making. One application of the MCS problem is in image segmentation, where the goal is to partition an image into regions based on color similarity. By formulating the segmentation task as an MCS problem on a graph representing pixel connectivity, one can efficiently identify consistent subsets of pixels with similar color attributes, leading to accurate segmentation results. Additionally, in social network analysis, the MCS problem can be used for community detection, where nodes with similar attributes or connections are grouped together to reveal underlying community structures. Insights from this work can be leveraged to develop algorithms for optimizing resource allocation, identifying cohesive groups in networks, and enhancing classification tasks in various domains. By leveraging the computational complexity results and algorithmic techniques developed for the MCS problem, researchers and practitioners can improve the efficiency and effectiveness of decision-making processes in diverse applications.