Computational Complexity of Finding Stable Sets in Graphs with Matroid Constraints

核心概念

The Independent Stable Set problem seeks to find a stable set of vertices in a graph that is also independent with respect to a given matroid. This problem generalizes several well-studied algorithmic problems, including Rainbow Independent Set, Rainbow Matching, and Bipartite Matching with Separation.

摘要

The paper studies the computational complexity of the Independent Stable Set problem. It provides the following key insights:

Unconditional lower bound: When the input matroids are represented by independence oracles, there is no algorithm that can solve Independent Stable Set using f(k) · no(k) oracle calls for any computable function f. This lower bound holds even for bipartite, chordal, claw-free, and AT-free graphs.

Parameterized complexity on sparse graphs:

For d-degenerate graphs, Independent Stable Set is FPT when parameterized by d + k, and admits a polynomial kernel when d is a constant.
However, the problem does not admit a polynomial kernel when parameterized by k + d unless NP ⊆ coNP/poly, even for partition matroids.
For graphs with maximum degree ∆, Independent Stable Set admits a polynomial kernel with a graph of size at most k^2∆.

Chordal graphs and linear matroids:

When the input graph is chordal and the matroid is linear, given by its representation, Independent Stable Set can be solved in 2^O(k) · ||A||^O(1) time by a one-sided error Monte Carlo algorithm.
However, the problem does not admit a polynomial kernel on chordal graphs and partition matroids, unless NP ⊆ coNP/poly.

The paper provides a comprehensive analysis of the computational complexity of Independent Stable Set, establishing both positive and negative results for various graph classes and matroid representations.

統計資料

None.

引述

None.

從以下內容提煉的關鍵洞見

Stability in Graphs with Matroid Constraints

by Fedor V. Fom... 於 arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03979.pdf

深入探究

How can the techniques developed in this paper be extended to other generalizations of the stable set problem, such as finding maximum weighted independent sets in frameworks

The techniques developed in the paper for Independent Stable Set can be extended to finding maximum weighted independent sets in frameworks by incorporating the concept of weights into the matroid representation. In the context of matroids, weights can be assigned to the elements of the ground set, and the rank function of the matroid can be defined in a way that considers these weights. This would involve optimizing the total weight of the independent set while ensuring that it satisfies the independence properties of the matroid.
The recursive branching algorithm used in the paper for Independent Stable Set on d-degenerate graphs can be adapted for finding maximum weighted independent sets. By modifying the branching criteria to consider the weights of the elements, the algorithm can explore different combinations of elements to maximize the total weight of the independent set. Additionally, the kernelization techniques can be applied to reduce the size of the input instance while preserving the weighted properties of the independent sets.

Are there other natural graph classes or matroid representations for which the complexity of Independent Stable Set can be further characterized

The complexity of Independent Stable Set can be further characterized for graph classes such as planar graphs, interval graphs, and cographs. For planar graphs, which have a variety of interesting structural properties, the algorithmic techniques developed in the paper may need to be adapted to account for the planar embedding and the restrictions it imposes on the graph structure. Interval graphs, which have a simple intersection model, could provide insights into efficient algorithms for Independent Stable Set based on interval representations.
Moreover, exploring the complexity of Independent Stable Set on cographs, which are recursively defined graphs with simple structural properties, could lead to a better understanding of the problem's computational complexity on graph classes with specific characteristics. Additionally, investigating different types of matroid representations, such as graphic matroids or transversal matroids, may offer new insights into the algorithmic complexity of Independent Stable Set.

What are the practical implications of these results, and how can they inform the design of algorithms for real-world applications involving stable sets and matroids

The results presented in the paper have practical implications for algorithm design in various real-world applications involving stable sets and matroids. For example, in scheduling problems where tasks have dependencies that form a matroid structure, the algorithms developed for Independent Stable Set can be used to optimize task assignments while respecting the dependencies. This can lead to more efficient scheduling and resource allocation in project management and production planning.
Furthermore, in wireless communication networks where interference constraints form a matroid, the techniques from the paper can be applied to find independent sets of communication channels that minimize interference and maximize network throughput. This can improve the overall performance and reliability of wireless networks. Overall, the results can inform the design of efficient algorithms for a wide range of optimization problems in various domains where stable sets and matroids play a crucial role.