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Bipartite Perfect Matching in Pseudo-Deterministic NC

Authors Shafi Goldwasser, Ofer Grossman

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Shafi Goldwasser
Ofer Grossman

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Shafi Goldwasser and Ofer Grossman. Bipartite Perfect Matching in Pseudo-Deterministic NC. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 87:1-87:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.87

Abstract

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph.

We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art.

Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm.

Subject Classification

Keywords
  • Parallel Algorithms
  • Pseudo-determinism
  • RNC
  • Perfect Matching

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References

  1. Alok Aggarwal, Richard J. Anderson, and M.-Y. Kao. Parallel depth-first search in general directed graphs. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 297-308. ACM, 1989. Google Scholar
  2. Noga Alon, Shlomo Hoory, and Nathan Linial. The Moore bound for irregular graphs. Graphs and Combinatorics, 18(1):53-57, 2002. Google Scholar
  3. Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Randomness-optimal unique element isolation with applications to perfect matching and related problems. SIAM Journal on Computing, 24(5):1036-1050, 1995. Google Scholar
  4. Samir Datta, Raghav Kulkarni, and Sambuddha Roy. Deterministically isolating a perfect matching in bipartite planar graphs. Theory of Computing Systems, 47(3):737-757, 2010. Google Scholar
  5. Bart de Smit and Hendrik W. Lenstra. Standard models for finite fields. In Handbook of finite fields, Discrete Mathematics and Its Applications. CRC Press, Hoboken, NJ, 2013. Google Scholar
  6. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17(3):449-467, 1965. Google Scholar
  7. Stephen Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite perfect matching is in quasi-NC. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 754-763, New York, NY, USA, 2016. ACM. URL: http://dx.doi.org/10.1145/2897518.2897564.
  8. Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. In Electronic Colloquium on Computational Complexity (ECCC), volume 18, page 136, 2011. Google Scholar
  9. Mohsen Ghaffari. Near-optimal scheduling of distributed algorithms. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pages 3-12. ACM, 2015. Google Scholar
  10. Oded Goldreich. In a world of P= BPP. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 191-232. Springer, 2011. Google Scholar
  11. Oded Goldreich, Shafi Goldwasser, and Dana Ron. On the possibilities and limitations of pseudodeterministic algorithms. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 127-138. ACM, 2013. Google Scholar
  12. Ofer Grossman. Finding primitive roots pseudo-deterministically. ECCC, 23rd December 2015. URL: http://eccc.hpi-web.de/report/2015/207/.
  13. Howard J. Karloff. A Las Vegas RNC algorithm for maximum matching. Combinatorica, 6(4):387-391, 1986. Google Scholar
  14. Richard M Karp, Eli Upfal, and Avi Wigderson. Constructing a perfect matching is in random NC. In Proceedings of the seventeenth annual ACM symposium on Theory of computing, pages 22-32. ACM, 1985. Google Scholar
  15. László Lovász. On determinants, matchings, and random algorithms. In FCT, volume 79, pages 565-574, 1979. Google Scholar
  16. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL: http://dx.doi.org/10.1007/BF02579206.
  17. Maria Serna and Paul Spirakis. Tight RNC approximations to max flow. In STACS 91, pages 118-126. Springer, 1991. Google Scholar
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