Efficient Algorithms for Computing Maximum Cliques in Disk and Ball Graphs with Bounded Radii Diversity

Concetti Chiave

This paper presents efficient algorithms for computing maximum cliques in disk graphs with a bounded number of different radii sizes, and in ball graphs with ball centers on a bounded number of parallel planes or planes perpendicular to a single plane.

Sintesi

The paper examines the problem of efficiently computing maximum cliques in disk graphs and ball graphs, which are geometric intersection graphs with disks and balls as vertices, respectively.

Key highlights:

• For disk graphs with k different radii sizes, the paper presents an O(2^k n^2k(f(n) + n^2))-time algorithm, where f(n) is the time to compute a maximum matching in a n-vertex bipartite graph. This settles the open question for the case of k=2 different radii.
• For unit disk graphs, the paper shows how to compute a maximum clique for every possible axis-aligned rectangle determined by the input disk centers in O(n^5 log n) time, which is at least a factor of n^4/3 faster than applying the fastest known algorithm for each rectangle independently.
• For ball graphs with k different radii sizes where the ball centers lie on r parallel planes, the paper gives an O(2^k n^2rk(f(n) + n^2r))-time algorithm. This contrasts the previously known NP-hardness result for finding a maximum clique in an arbitrary ball graph.
• The key ideas behind the algorithms are to exploit the geometric properties of disks and balls to efficiently construct cliques, rather than relying on the traditional lens-based approach which faces challenges in the multi-radii setting.

Statistiche

The paper does not provide any specific numerical data or statistics to support the key logics. The focus is on presenting efficient algorithms and their time complexities.

Citazioni

The paper does not contain any striking quotes that support the key logics.

Approfondimenti chiave tratti da

The Maximum Clique Problem in a Disk Graph Made Easy

by J. Mark Keil... alle arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03751.pdf

Domande più approfondite

How can the time complexities of the presented algorithms be further improved, especially for the case when the number of different radii sizes is not constant

To improve the time complexities of the algorithms for cases with varying numbers of different radii sizes, several strategies can be considered. One approach could involve optimizing the data structures used for storing and processing the information about the disks or balls. By designing more efficient data structures tailored to the specific characteristics of the problem, the algorithm's performance could be enhanced. Additionally, exploring parallelization techniques to leverage multiple processors or cores could help reduce the overall computation time. By dividing the workload and processing different parts simultaneously, the algorithm's efficiency could be significantly improved. Furthermore, refining the algorithm's logic and optimizing the computational steps could lead to a more streamlined and faster solution. By carefully analyzing the algorithm's flow and identifying areas for optimization, it may be possible to reduce unnecessary computations and streamline the overall process, resulting in improved time complexities.

Can the perpendicularity constraint on the input planes be removed for the ball graph case while still maintaining polynomial-time algorithms

Removing the perpendicularity constraint on the input planes for the ball graph case while maintaining polynomial-time algorithms could be a challenging task but is certainly worth exploring. One potential approach could involve transforming the input data to a different coordinate system where the planes are no longer perpendicular. By applying appropriate transformations and adjustments to the input data, it may be possible to adapt the existing algorithms to handle non-perpendicular planes while still achieving polynomial-time complexity. Additionally, exploring alternative algorithmic techniques that are more flexible and adaptable to different plane configurations could provide insights into how to address this constraint. By investigating the underlying principles of the algorithms and identifying the key dependencies on the perpendicularity constraint, it may be possible to devise new strategies that can handle a broader range of input configurations.

What are the implications of these results on other geometric intersection graph problems beyond the maximum clique problem

The results presented in the context of geometric intersection graph problems, particularly the maximum clique problem in disk and ball graphs, have broader implications for various other computational geometry problems. The development of efficient algorithms for these specific graph classes opens up possibilities for addressing similar problems in different geometric contexts. For instance, the techniques and insights gained from solving the maximum clique problem in disk and ball graphs could be applied to other intersection graph problems, such as unit disk graphs or trapezoid graphs. By leveraging the algorithmic strategies and data structures developed for these specific cases, researchers can potentially tackle a wide range of geometric intersection graph problems more effectively. Additionally, the results may inspire further research into optimizing algorithms for different geometric graph classes and exploring new applications in areas such as wireless networks, VLSI design, and computational geometry.