Bibliographic Information: Hébert-Johnson, Ú., Lokshtanov, D., & Vigoda, E. (2024). Counting and Sampling Labeled Chordal Graphs in Polynomial Time. arXiv preprint arXiv:2308.09703v2.
Research Objective: To develop a polynomial-time algorithm for counting and uniformly sampling labeled chordal graphs.
Methodology: The researchers designed a dynamic programming algorithm that leverages the concept of "evaporation sequences," a canonical representation of perfect elimination orderings in chordal graphs. This approach allows for the systematic enumeration of all labeled chordal graphs with a given number of vertices. The counting algorithm is then extended to a uniform sampling algorithm using a standard sampling-to-counting reduction technique.
Key Findings:
Main Conclusions: The authors successfully addressed a long-standing open problem in graph theory by providing an efficient method for counting and sampling labeled chordal graphs. This breakthrough has implications for various applications, including graph generation, simulation studies, and algorithm testing within the domain of chordal graphs.
Significance: This research significantly advances the field of graph algorithms by providing a practical solution to a computationally challenging problem. The ability to efficiently count and sample chordal graphs opens up new avenues for research and applications involving this important graph class.
Limitations and Future Research: While the polynomial-time algorithm is a major breakthrough, its O(n^7) running time might still be computationally demanding for very large graphs. Future research could explore optimizations to further reduce the runtime complexity. Additionally, investigating efficient algorithms for counting and sampling labeled chordal graphs with specific properties (e.g., bounded degree) remains an open area for exploration.
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by Ursula Heber... at arxiv.org 10-04-2024
https://arxiv.org/pdf/2308.09703.pdfDeeper Inquiries