An FPRAS for Two Terminal Reliability in Directed Acyclic Graphs

Authors Weiming Feng , Heng Guo



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Author Details

Weiming Feng
  • Institute for Theoretical Studies, ETH Zürich, Switzerland
Heng Guo
  • School of Informatics, University of Edinburgh, UK

Acknowledgements

We thank Kuldeep S. Meel for bringing the problem to our attention, Antoine Amarilli for explaining their method to us, and Marcelo Arenas for insightful discussions. We also thank Zongchen Chen for suggesting a better presentation of Theorem 1, and Mark Jerrum for some useful insights. This work was done in part while Weiming Feng was visiting the Simons Institute for the Theory of Computing.

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Weiming Feng and Heng Guo. An FPRAS for Two Terminal Reliability in Directed Acyclic Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.62

Abstract

We give a fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs). In contrast, we also show the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.

Subject Classification

ACM Subject Classification
  • Networks → Network reliability
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Generating random combinatorial structures
Keywords
  • Approximate counting
  • Network reliability
  • Sampling algorithm

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References

  1. Carme Àlvarez and Birgit Jenner. A very hard log-space counting class. Theor. Comput. Sci., 107(1):3-30, 1993. Google Scholar
  2. Antoine Amarilli, Timothy van Bremen, and Kuldeep S. Meel. Conjunctive queries on probabilistic graphs: the limits of approximability. arXiv, abs/2309.13287, 2023. URL: https://arxiv.org/abs/2309.13287.
  3. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In STOC, pages 1-12. ACM, 2019. Google Scholar
  4. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant, and Thuy-Duong Vuong. Log-concave polynomials IV: approximate exchange, tight mixing times, and near-optimal sampling of forests. In STOC, pages 408-420. ACM, 2021. Google Scholar
  5. Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. #NFA admits an FPRAS: efficient enumeration, counting, and uniform generation for logspace classes. J. ACM, 68(6):48:1-48:40, 2021. Google Scholar
  6. Michael O. Ball. Complexity of network reliability computations. Networks, 10(2):153-165, 1980. Google Scholar
  7. Michael O. Ball. Computational complexity of network reliability analysis: An overview. IEEE Trans. Rel., 35(3):230-239, 1986. Google Scholar
  8. Michael O. Ball and J. Scott Provan. Calculating bounds on reachability and connectedness in stochastic networks. Networks, 13(2):253-278, 1983. Google Scholar
  9. Ruoxu Cen, William He, Jason Li, and Debmalya Panigrahi. Beyond the quadratic time barrier for network unreliability. arXiv, abs/2304.06552, 2023. URL: https://arxiv.org/abs/2304.06552.
  10. Xiaoyu Chen, Heng Guo, Xinyuan Zhang, and Zongrui Zou. Near-linear time samplers for matroid independent sets with applications. arXiv, abs/2308.09683, 2023. URL: https://arxiv.org/abs/2308.09683.
  11. Charles J. Colbourn. The Combinatorics of Network Reliability. Oxford University Press, 1987. Google Scholar
  12. Mary Cryan, Heng Guo, and Giorgos Mousa. Modified log-Sobolev inequalities for strongly log-concave distributions. Ann. Probab., 49(1):506-525, 2021. Google Scholar
  13. Martin E. Dyer, Leslie Ann Goldberg, Catherine S. Greenhill, and Mark Jerrum. The relative complexity of approximate counting problems. Algorithmica, 38(3):471-500, 2004. Google Scholar
  14. Vivek Gore, Mark Jerrum, Sampath Kannan, Z. Sweedyk, and Stephen R. Mahaney. A quasi-polynomial-time algorithm for sampling words from a context-free language. Inf. Comput., 134(1):59-74, 1997. Google Scholar
  15. Heng Guo and Kun He. Tight bounds for popping algorithms. Random Struct. Algorithms, 57(2):371-392, 2020. Google Scholar
  16. Heng Guo and Mark Jerrum. A polynomial-time approximation algorithm for all-terminal network reliability. SIAM J. Comput., 48(3):964-978, 2019. Google Scholar
  17. Heng Guo, Mark Jerrum, and Jingcheng Liu. Uniform sampling through the Lovász local lemma. J. ACM, 66(3):18:1-18:31, 2019. Google Scholar
  18. David G. Harris and Aravind Srinivasan. Improved bounds and algorithms for graph cuts and network reliability. Random Struct. Algorithms, 52(1):74-135, 2018. Google Scholar
  19. Mark Jerrum. On the complexity of evaluating multivariate polynomials. PhD thesis, The University of Edinburgh, 1981. Google Scholar
  20. Mark Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci., 43:169-188, 1986. Google Scholar
  21. David R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM J. Comput., 29(2):492-514, 1999. Google Scholar
  22. David R. Karger. A phase transition and a quadratic time unbiased estimator for network reliability. In STOC, pages 485-495. ACM, 2020. Google Scholar
  23. Richard M. Karp and Michael Luby. Monte-Carlo algorithms for enumeration and reliability problems. In FOCS, pages 56-64. IEEE Computer Society, 1983. Google Scholar
  24. Richard M. Karp and Michael Luby. Monte-Carlo algorithms for the planar multiterminal network reliability problem. J. Complex., 1(1):45-64, 1985. Google Scholar
  25. Richard M. Karp, Michael Luby, and Neal Madras. Monte-Carlo approximation algorithms for enumeration problems. J. Algorithms, 10(3):429-448, 1989. Google Scholar
  26. Kuldeep S. Meel, Sourav Chakraborty, and Umang Mathur. A faster FPRAS for #NFA. arXiv, abs/2312.13320, 2023. URL: https://arxiv.org/abs/2312.13320.
  27. J. Scott Provan. The complexity of reliability computations in planar and acyclic graphs. SIAM J. Comput., 15(3):694-702, 1986. Google Scholar
  28. J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput., 12(4):777-788, 1983. Google Scholar
  29. Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM J. Comput., 8(3):410-421, 1979. Google Scholar
  30. Rico Zenklusen and Marco Laumanns. High-confidence estimation of small s-t reliabilities in directed acyclic networks. Networks, 57(4):376-388, 2011. Google Scholar