The Computational Complexity of Cycle Convexity Parameters in Graphs

Concepts de base

The computational complexity of determining the interval number, hull number, rank, and convexity number in the cycle convexity of graphs is analyzed, showing that these problems are NP-complete and W[1]-hard.

Résumé

The content explores the computational complexity of various parameters in the cycle convexity of graphs, which is a recently defined graph convexity concept.
Key highlights:

The interval number in cycle convexity can be computed in linear time for graphs with two universal vertices, but is NP-complete and W[2]-hard for bipartite graphs.
The hull number in cycle convexity is NP-complete even for simple planar graphs.
The rank in cycle convexity is NP-complete and W[1]-hard, as it is equivalent to the problem of finding the maximum induced forest.
The convexity number in cycle convexity is NP-complete and W[1]-hard.
The percolation time, which is the largest integer k such that there exists a hull set S with Ik-1(S) ≠ V, is NP-complete for any fixed k ≥ 9, but can be computed in cubic time for k = 2.
The article provides detailed proofs and analysis for establishing these computational complexity results, demonstrating the inherent difficulty of working with cycle convexity parameters.

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Idées clés tirées de

On the complexity of some cycle convexity parameters

by Carlos V.G.C... à arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09236.pdf

Questions plus approfondies

How do the computational complexity results for cycle convexity compare to those of other well-studied graph convexities, such as geodesic or P3 convexity

The computational complexity results for cycle convexity parameters, such as rank, convexity number, and percolation time, exhibit similarities and differences compared to other well-studied graph convexities like geodesic or P3 convexity. In terms of similarities, all these convexity parameters involve determining certain subsets of vertices that satisfy specific properties based on the graph structure. They all deal with the concept of infecting or covering vertices in a graph based on certain rules or constraints.
However, the complexity results for cycle convexity parameters can differ significantly from those of geodesic or P3 convexity. For example, the NP-completeness and hardness results for parameters like rank and convexity number in cycle convexity indicate that these problems are computationally challenging and belong to different complexity classes. In contrast, the complexity results for geodesic or P3 convexity may vary based on the specific constraints and rules governing those convexities.
Overall, while there may be some commonalities in the computational complexity of different graph convexities, the specific properties and rules defining each convexity parameter lead to distinct complexity results.

What are the practical implications of the hardness results for cycle convexity parameters, and how might they impact the design of algorithms and applications that rely on these concepts

The hardness results for cycle convexity parameters have significant practical implications for algorithm design and applications that rely on these concepts. The NP-completeness and hardness of problems associated with parameters like rank, convexity number, and percolation time in cycle convexity indicate that finding optimal solutions or determining certain properties of graphs within this context can be computationally intensive and challenging.
These hardness results can impact the development of algorithms for tasks such as graph analysis, network optimization, and infection modeling. Researchers and practitioners working in fields like social network analysis, epidemiology, and network security need to consider the computational complexity of cycle convexity parameters when designing algorithms and models. They may need to explore approximation algorithms, heuristic methods, or specialized techniques to handle the complexity of these problems efficiently.
Understanding the computational hardness of cycle convexity parameters can also guide the selection of appropriate data structures, optimization strategies, and algorithmic approaches to tackle these challenges effectively in real-world applications.

Are there any special graph classes or additional structural properties that could lead to more efficient algorithms for computing cycle convexity parameters

There are certain graph classes and structural properties that could potentially lead to more efficient algorithms for computing cycle convexity parameters. For example:

Tree-like Structures: Graphs with tree-like structures, such as cacti or forests, may exhibit specific properties that simplify the computation of cycle convexity parameters. The absence of cycles or the presence of limited cycle configurations in these graphs could lead to more straightforward algorithms for determining parameters like convexity number or percolation time.

Bipartite Graphs: Bipartite graphs have unique structural characteristics that could be leveraged to develop efficient algorithms for cycle convexity parameters. The bipartite nature of these graphs, with vertices divided into two disjoint sets, may allow for specialized algorithms that exploit this structure to compute convexity properties more effectively.

Planar Graphs: Planar graphs, which can be embedded in the plane without edge crossings, offer opportunities for algorithmic optimizations due to their inherent geometric properties. Algorithms tailored to exploit the planarity of these graphs could lead to more efficient computations of cycle convexity parameters.

By identifying and utilizing these special graph classes and structural properties, researchers and algorithm designers can explore tailored approaches to compute cycle convexity parameters efficiently in specific graph scenarios. These insights can contribute to the development of optimized algorithms for practical applications requiring the analysis of cycle convexity in graphs.