Approximation Ratio of Greedy Algorithm for Maximum Independent Set on Interval and Chordal Graphs

แนวคิดหลัก

Greedy algorithm is a (2/3)-approximation for Maximum Independent Set on interval graphs and a (1/2)-approximation on chordal graphs.

บทคัดย่อ

The article discusses the approximation ratio of the minimum-degree greedy algorithm for the Maximum Independent Set problem on interval and chordal graphs. It proves that the algorithm is a (2/3)-approximation for interval graphs, even on unit interval graphs of maximum degree 3, and a (1/2)-approximation for chordal graphs. The results contrast with the known approximation ratio of 3∆+2 for general graphs. The study focuses on worst-case tie-breaking scenarios to analyze how close the Greedy algorithm comes to computing a maximum independent set. It also delves into tree decompositions, simplicial vertices, and various observations related to Greedy's performance on different graph types.

สถิติ

Greedy is a (2/3)-approximation for Maximum Independent Set on interval graphs.
Greedy is a (1/2)-approximation for Maximum Independent Set on chordal graphs.
Approximation ratio of 3∆+2 for general graphs.

คำพูด

"The minimum-degree greedy algorithm is a (2/3)-approximation for the Maximum Independent Set problem on interval graphs."
"Greedy is a (1/2)-approximation algorithm on chordal graphs."

ข้อมูลเชิงลึกที่สำคัญจาก

Approximation Ratio of the Min-Degree Greedy Algorithm for Maximum Independent Set on Interval and Chordal Graphs

by Steven Chapl... ที่ arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.10868.pdf

สอบถามเพิ่มเติม

How does the performance of the Greedy algorithm compare to other approximation algorithms

The performance of the Greedy algorithm in terms of approximation ratios varies depending on the type of graph being analyzed. In the context provided, it is shown that on interval graphs, the Greedy algorithm with adversarial tie-breaking achieves a (2/3)-approximation for the Maximum Independent Set problem. On chordal graphs, this ratio improves to a (1/2)-approximation. These results contrast with the known tight approximation ratio of 3∆+2 for general graphs of maximum degree ∆.

What implications do these findings have for real-world applications of graph theory

The findings regarding the performance of the Greedy algorithm on interval and chordal graphs have significant implications for real-world applications of graph theory. For instance, in scheduling problems where tasks can be represented as intervals (as seen in interval scheduling literature), understanding that Greedy provides a (2/3)-approximation on interval graphs can help optimize task allocation and resource management processes efficiently. Similarly, in network design or optimization scenarios where structures exhibit chordal properties, knowing that Greedy offers a tighter (1/2)-approximation can lead to more effective decision-making strategies.

How can tie-breaking strategies impact the efficiency of approximation algorithms

Tie-breaking strategies play a crucial role in determining the efficiency and effectiveness of approximation algorithms like Greedy. In cases where ties occur during vertex selection, how these ties are resolved can impact both the final solution quality and computational complexity. The choice between adversarial tie-breaking and advantageous tie-breaking methods can significantly influence whether an algorithm achieves its desired approximation guarantee or not. By studying worst-case scenarios with adversarial tie-breaking as done in this analysis, researchers gain insights into how robust an algorithm's performance is under challenging conditions and provide valuable guidance for practical implementations where optimal solutions may not always be feasible.