Document

# Sequential Importance Sampling Algorithms for Estimating the All-Terminal Reliability Polynomial of Sparse Graphs

## File

LIPIcs.APPROX-RANDOM.2015.323.pdf
• Filesize: 482 kB
• 18 pages

## Cite As

David G. Harris and Francis Sullivan. Sequential Importance Sampling Algorithms for Estimating the All-Terminal Reliability Polynomial of Sparse Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 323-340, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.323

## Abstract

The all-terminal reliability polynomial of a graph counts its connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph's reliability polynomial. We show upper bounds on the relative error of three sequential importance sampling algorithms. We use these to create a hybrid algorithm, which selects the best SIS algorithm for a particular graph G and particular coefficient of the polynomial. This hybrid algorithm is particularly effective when G has low degree. For graphs of average degree < 11, it is the fastest known algorithm; for graphs of average degree <= 45 it is the fastest known polynomial-space algorithm. For example, when a graph has average degree 3, this algorithm estimates to error epsilon in time O(1.26^n * epsilon^{-2}). Although the algorithm may take exponential time, in practice it can have good performance even on medium-scale graphs. We provide experimental results that show quite practical performance on graphs with hundreds of vertices and thousands of edges. By contrast, alternative algorithms are either not rigorous or are completely impractical for such large graphs.
##### Keywords
• All-terminal reliability
• sequential importance sampling

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Michael O. Ball and J. Scott Provan. Bounds on the reliablity polynomial for shellable independence systems. SIAM Journal on Algebraic and Discrete Methods, 3:166-181, 1982.
2. Isabel Beichl, Brian Cloteaux, and Francis Sullivan. An approximation algorithm for the coefficients of the reliability polynomial. Congressus Numerantium, 197:143-151, 2010.
3. Andreas Bjorklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Computing the Tutte polynomial in vertex-exponential time. Foundations of Computer Science (FOCS), pages 677-686, 2008.
4. Adams L. Buchsbaum and Milena Mihail. Monte Carlo and Markov chain techniques for network reliability and sampling. Networks, 25:117-130, 1995.
5. Shiri Chechik, Yuval Emek, Boaz Patt-Shamir, and David Peleg. Sparse reliable graph backbones. Automata, Languages, and Processing, pages 261-272, 2010.
6. Andrew Chen. On graphs with large numbers of spanning trees. PhD dissertation for Michigan State University Department of Computer Science, 2005.
7. Fan RK Chung and Ronald L. Graham. On the cover polynomial of a digraph. Journal of Combinatorial Theory, Series B, 65:273-290, 1995.
8. Charles J. Colbourn, Bradley M. Debroni, and Wendy J. Myrold. Estimating the coefficients of the reliability polynomial. Congress Numerantium, 62:217-223, 1988.
9. Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlén. Exponential time complexity of the permanent and Tutte polynomial. ACM Transactions on Algorithms, 10-4, 2014.
10. George S. Fishman. A Monte Carlo sampling plan for estimating network reliability. Operations Research, 34:581-594, 1986.
11. Heidi Gebauer and Yoshio Okamoto. Fast exponential-time algorithms for the forest counting in graph classes. Theory of Computing Australasian Symposium, 65:63-69, 2007.
12. Leslie Ann Goldberg and Mark Jerrum. Inapproximability of the Tutte polynomial. Information and Computation, 206-7:908-929, 2008.
13. Gary Haggard, David J. Pearce, and Gordon Royle. Computing Tutte polynomials. ACM Transactions on Mathematical Software, 37-3 Article # 24, 2010.
14. David G. Harris, Francis Sullivan, and Isabel Beichl. Linear algebra and sequential importance sampling for network reliability. Winter Simulation Conference, 2011.
15. David G. Harris, Francis Sullivan, and Isabel Beichl. Fast sequential importance sampling to estimate the graph reliability polynomial. Algorithmica, 68-4:916-939, 2014.
16. David Karger. A randomized fully polynomial time approximation scheme for the all terminal network reliability problem. SIAM Journal on Computing, 29:11-17, 1996.
17. Joseph B. Kruskal. The number of simplices in a complex. Mathematical Optimization Techniques, 1963.
18. Marco Laumanns and Rico Zenklusen. High-confidence estimation of small s-t reliabilities in directed acylic networks. Networks, 57-4:367-388, 2011.
19. Wendy Myrvold. Counting k-component forests of a graph. Networks, 22:647-652, 1992.
20. J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing, 12:777-788, 1983.
21. Kyoko Sekine, Hiroshi Imai, and Seiichiro Tani. Computing the Tutte polynomial of a graph of moderate size. Lecture Notes in Computer Science, 1004:224-233, 1995.
22. David Bruce Wilson. Generating random spanning trees more quickly than the cover time. ACM Symposium on Theory of Computing (STOC), pages 296-303, 1996.
X

Feedback for Dagstuhl Publishing