Core Concepts
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.
Abstract
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.
Stats
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.
Quotes
The paper does not contain any striking quotes that support the key logics.