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Reachability in O(log n) Genus Graphs is in Unambiguous Logspace

Authors Chetan Gupta, Vimal Raj Sharma, Raghunath Tewari



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Author Details

Chetan Gupta
  • Indian Institute of Technology, Kanpur, India
Vimal Raj Sharma
  • Indian Institute of Technology, Kanpur, India
Raghunath Tewari
  • Indian Institute of Technology, Kanpur, India

Acknowledgements

The authors would like to thank Ministry of Electronics and IT, India for supporting this research through the Visvesvaraya PhD and YFRF program. The third author would also like to thank DST for providing funding support.

Cite AsGet BibTex

Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Reachability in O(log n) Genus Graphs is in Unambiguous Logspace. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 34:1-34:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.34

Abstract

We show that given an embedding of an O(log n) genus graph G and two vertices s and t in G, deciding if there is a path from s to t in G is in unambiguous logarithmic space. Unambiguous computation is a restriction of nondeterministic computation where the nondeterministic machine has at most one accepting computation path on each input. An important fundamental question in computational complexity theory is whether this is an actual restriction or are unambiguous computations as powerful as general nondeterminism. We investigate this problem in the domain of logarithmic space bounded computations, where the corresponding unambiguous and general nondeterministic classes are UL and NL respectively. In 1997 Reinhardt and Allender showed that NL and UL are equal in a non-uniform model. More specifically they showed that if one can efficiently construct an O(log n)-bit min-unique weight function for a graph, then these classes are equal unconditionally as well. In other words, they gave a UL algorithm to solve reachability in graphs with a min-unique weight assignment. Using this approach reachability in various classes of graphs such as planar graphs, constant genus graphs, minor free graphs, etc., have been shown to be in UL by devising min-unique weight functions for those classes. In this paper we improve these results by constructing a min-unique weight function for O(log n) genus graphs. We define signature of a path in a graph as the parity of the number of crossings of that path with respect to each handle of the surface on which the graph is embedded. We construct our weight function in two steps. First we ensure that between any pair of vertices, amongst all paths having the same signature, the minimum weight path is unique. Now since in a genus g graph there are 2^{2g} many possible signatures, we use the hashing scheme of Fredman, Komlós and Szemerédi to isolate a unique minimum weight path among these 2^{2g} many paths isolated in the first step.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • logspace unambiguity
  • high genus
  • path isolation

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References

  1. Manindra Agrawal, Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits. SIAM J. Comput., 44(3):669-697, 2015. URL: http://dx.doi.org/10.1137/140975103.
  2. Eric Allender, David A. Mix Barrington, Tanmoy Chakraborty, Samir Datta, and Sambuddha Roy. Planar and Grid Graph Reachability Problems. Theory of Computing Systems, 45(4):675-723, 2009. URL: http://dx.doi.org/10.1007/s00224-009-9172-z.
  3. Eric Allender, Klaus Reinhardt, and Shiyu Zhou. Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds. Journal of Computer and System Sciences, 59:164-181, 1999. Google Scholar
  4. Carme Àlvarez and Birgit Jenner. A Very Hard Log-space Counting Class. Theoretical Computer Science, 107:3-30, 1993. Google Scholar
  5. Rahul Arora, Ashu Gupta, Rohit Gurjar, and Raghunath Tewari. Derandomizing Isolation Lemma for K_3, 3-free and K₅-free Bipartite Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 10:1-10:15, 2016. Google Scholar
  6. David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC¹. Journal of Computer and System Sciences, 38:150-164, 1989. Google Scholar
  7. David A. Mix Barrington, Chi-Jen Lu, Peter Bro Miltersen, and Sven Skyum. Searching constant width mazes captures the ∿⁰ hierarchy. In 15th International Symposium on Theoretical Aspects of Computer Science (STACS), Volume 1373 in Lecture Notes in Computer Science, pages 74-83. Springer, 1998. Google Scholar
  8. Chris Bourke, Raghunath Tewari, and N. V. Vinodchandran. Directed Planar Reachability Is in Unambiguous Log-Space. ACM Transactions on Computation Theory, 1(1):1-17, 2009. URL: http://dx.doi.org/10.1145/1490270.1490274.
  9. Gerhard Buntrock, Birgit Jenner, Klaus-Jörn Lange, and Peter Rossmanith. Unambiguity and fewness for logarithmic space. In Proceedings of the 8th International Conference on Fundamentals of Computation Theory (FCT'91), Volume 529 Lecture Notes in Computer Science, pages 168-179. Springer-Verlag, 1991. Google Scholar
  10. Sergio Cabello and Bojan Mohar. Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs. Discrete & Computational Geometry, 37(2):213-235, February 2007. URL: http://dx.doi.org/10.1007/s00454-006-1292-5.
  11. Samir Datta, Raghav Kulkarni, and Sambuddha Roy. Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs. In STACS 2008, 25th Annual Symposium on Theoretical Aspects of Computer Science, Bordeaux, France, February 21-23, 2008, Proceedings, pages 229-240, 2008. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1346.
  12. Samir Datta, Raghav Kulkarni, Raghunath Tewari, and N.V. Vinodchandran. Space complexity of perfect matching in bounded genus bipartite graphs. Journal of Computer and System Sciences, 78(3):765-779, 2012. In Commemoration of Amir Pnueli. URL: http://dx.doi.org/10.1016/j.jcss.2011.11.002.
  13. Kousha Etessami. Counting quantifiers, successor relations, and logarithmic space. Journal of Computer and System Sciences, 54(3):400-411, June 1997. Google Scholar
  14. Stephen A. Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite Perfect Matching is in quasi-NC. CoRR, 2016. URL: http://arxiv.org/abs/1601.06319.
  15. Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a Sparse Table with 0(1) Worst Case Access Time. J. ACM, 31(3):538-544, June 1984. URL: http://dx.doi.org/10.1145/828.1884.
  16. Anna Gál and Avi Wigderson. Boolean complexity classes vs. their arithmetic analogs. Random Struct. Algorithms, 9(1-2):99-111, 1996. URL: http://dx.doi.org/10.1002/(SICI)1098-2418(199608/09)9:1/2%3C99::AID-RSA7%3E3.0.CO;2-6.
  17. Anna Galluccio and Martin Loebl. On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents. Electr. J. Comb., 6, 1999. URL: http://www.combinatorics.org/Volume_6/Abstracts/v6i1r6.html.
  18. Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 29:1-29:16, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.29.
  19. Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs. In Proceedings of the 30th Conference on Computational Complexity, CCC '15, pages 323-346, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.323.
  20. Vivek Anand T. Kallampally and Raghunath Tewari. Trading Determinism for Time in Space Bounded Computations. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 10:1-10:13, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.10.
  21. Jan Kynčl and Tomáš Vyskočil. Logspace Reduction of Directed Reachability for Bounded Genus Graphs to the Planar Case. ACM Transactions on Computation Theory, 1(3):1-11, 2010. URL: http://dx.doi.org/10.1145/1714450.1714451.
  22. Meena Mahajan and Kasturi R. Varadarajan. A New NC-algorithm for Finding a Perfect Matching in Bipartite Planar and Small Genus Graphs (Extended Abstract). In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, STOC '00, pages 351-357, New York, NY, USA, 2000. ACM. URL: http://dx.doi.org/10.1145/335305.335346.
  23. Aduri Pavan, Raghunath Tewari, and N. V. Vinodchandran. On the power of unambiguity in log-space. Computational Complexity, 21(4):643-670, 2012. URL: http://dx.doi.org/10.1007/s00037-012-0047-3.
  24. Omer Reingold. Undirected connectivity in log-space. Journal of the ACM, 55(4), 2008. URL: http://dx.doi.org/10.1145/1391289.1391291.
  25. Klaus Reinhardt and Eric Allender. Making Nondeterminism Unambiguous. SIAM J. Comput., 29(4):1118-1131, 2000. URL: http://dx.doi.org/10.1137/S0097539798339041.
  26. Raghunath Tewari and N. V. Vinodchandran. Green’s theorem and isolation in planar graphs. Inf. Comput., 215:1-7, 2012. URL: http://dx.doi.org/10.1016/j.ic.2012.03.002.
  27. Thomas Thierauf and Fabian Wagner. Reachability in K_3,3-free Graphs and K₅-free Graphs is in Unambiguous Log-Space. In 17th International Conference on Foundations of Computation Theory (FCT), Lecture Notes in Computer Science 5699, pages 323-334. Springer-Verlag, 2009. Google Scholar
  28. Carsten Thomassen. The Graph Genus Problem is NP-Complete. J. Algorithms, 10(4):568-576, 1989. URL: http://dx.doi.org/10.1016/0196-6774(89)90006-0.
  29. Dieter van Melkebeek and Gautam Prakriya. Derandomizing Isolation in Space-Bounded Settings. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 5:1-5:32, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.5.
  30. Avi Wigderson. NL/poly ⊆ ⊕ L/poly (Preliminary Version). In Proceedings of the Ninth Annual Structure in Complexity Theory Conference, Amsterdam, The Netherlands, June 28 - July 1, 1994, pages 59-62, 1994. URL: http://dx.doi.org/10.1109/SCT.1994.315817.
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