Additive Approximation Algorithm for k-Geodesic Center in δ-Hyperbolic Graphs

แนวคิดหลัก

The authors provide an additive O(δ)-approximation algorithm for the k-Geodesic Center problem on δ-hyperbolic graphs, where the goal is to find a set of k isometric paths that minimizes the maximum distance between any vertex and the set of paths.

บทคัดย่อ

The key highlights and insights of the content are:

1. The k-Geodesic Center problem asks to find a collection C of k isometric paths such that the maximum distance between any vertex and C is minimized. This problem is a generalization of the Minimum Eccentricity Shortest Path (MESP) problem.

2. The authors introduce the notion of a "shallow pairing" property, which is a coarse version of the pairing property introduced by Gerstel & Zaks. They show that δ-hyperbolic graphs satisfy the (2δ + 1/2)-shallow pairing property.

3. The authors provide a two-stage algorithm for k-Geodesic Center on δ-hyperbolic graphs:

   • In the first stage, they solve the "rooted" version of the (2k-1)-Geodesic Center problem, where all isometric paths in the solution have a common end-vertex.
   • In the second stage, they use the shallow pairing property to transform the rooted solution into a non-rooted solution of size k, while incurring an additive O(δ) error.

4. The authors also adapt a technique from Dragan & Leitert to show that the k-Geodesic Center problem is NP-hard even on partial grids (subgraphs of (k × k)-grids).

5. For the special case of trees, the authors' algorithmic approach leads to an exact polynomial-time algorithm for the k-Geodesic Center problem.

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ข้อมูลเชิงลึกที่สำคัญจาก

Additive approximation algorithm for geodesic centers in $δ$-hyperbolic graphs

by Diby... ที่ arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03812.pdf

สอบถามเพิ่มเติม

What are some real-world applications of the k-Geodesic Center problem beyond the examples mentioned in the content (communication networks, transportation planning, water resource management, fluid transportation)

The k-Geodesic Center problem has various real-world applications beyond the ones mentioned in the context. One such application is in urban planning and city infrastructure development. By identifying the k-Geodesic Centers in a city's road network, urban planners can optimize the placement of essential services such as hospitals, fire stations, and police stations to ensure efficient coverage and accessibility to the entire population. Additionally, in social network analysis, identifying k-Geodesic Centers can help in understanding the flow of information, influence, or trends within a network, aiding in targeted marketing strategies or content dissemination. Furthermore, in biological networks, such as protein-protein interaction networks, identifying k-Geodesic Centers can help in understanding key proteins or genes that play crucial roles in various biological processes.

How does the performance of the proposed additive approximation algorithm compare to other known approximation algorithms for the k-Geodesic Center problem on general graphs

The proposed additive approximation algorithm for the k-Geodesic Center problem on δ-hyperbolic graphs provides a significant improvement in terms of performance compared to other known approximation algorithms. The algorithm offers an additive O(δ)-approximation for k-Geodesic Center, which is a notable achievement in terms of accuracy and efficiency. This algorithm's performance is particularly noteworthy in the context of δ-hyperbolic graphs, where the algorithm's complexity is polynomial and depends only on the δ-hyperbolicity of the input graph. This makes it a valuable tool for practical applications where efficiency and accuracy are essential.

Can the techniques used in this paper be extended to solve other related problems, such as the Isometric Path Cover problem, on δ-hyperbolic graphs

The techniques used in this paper to solve the k-Geodesic Center problem on δ-hyperbolic graphs can potentially be extended to address other related problems, such as the Isometric Path Cover problem. By adapting the algorithmic approach and concepts introduced in this paper, it may be possible to develop additive approximation algorithms for Isometric Path Cover on δ-hyperbolic graphs. The underlying principles of rooted counterparts, shallow pairing properties, and the reduction of problems to rooted versions can be applied to formulate solutions for similar graph optimization problems in the context of δ-hyperbolicity. This extension could contribute to advancing the algorithmic study of graph optimization problems on δ-hyperbolic graphs.