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Complexity Dichotomy for Counting Problems

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Language and Automata Theory and Applications (LATA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7810))

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Abstract

I would like to report on some significant progress in the study of the exact complexity of counting problems. Specifically I will describe the classification program of counting complexity of locally specified problems. This classification program is advanced in three interrelated frameworks: Graph Homomorphisms, Counting CSP, and Holant Problems. In each formulation, complexity dichotomy theorems have been achieved which classify every problem in a given class to be either solvable in polynomial time or #P-hard.

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References

  1. Bulatov, A.A.: The Complexity of the Counting Constraint Satisfaction Problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 646–661. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348(2), 148–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J.-Y., Chen, X.: A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 437–446 (2010)

    Google Scholar 

  4. Cai, J.-Y., Chen, X.: Complexity of counting CSP with complex weights. In: Proceedings of the 44th Symposium on Theory of Computing (STOC 2012), pp. 909–920 (2012), http://arxiv.org/abs/1111.2384

  5. Cai, J.-Y., Chen, X., Lu, P.: Graph Homomorphisms with Complex Values: A Dichotomy Theorem. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 275–286. Springer, Heidelberg (2010), http://arxiv.org/abs/0903.4728

    Chapter  Google Scholar 

  6. Cai, J.-Y., Guo, H., Williams, T.: A Complete Dichotomy Rises from the Capture of Vanishing Signatures, http://arxiv.org/abs/1204.6445

  7. Cai, J.-Y., Lu, P.: Holographic algorithms: from art to science. In: STOC 2007: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 401–410. ACM, New York (2007); Journal of Computer and System Sciences 77(1), 41–61 (2011)

    Chapter  Google Scholar 

  8. Cai, J.-Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 715–724 (2009)

    Google Scholar 

  9. Cai, J.-Y., Lu, P., Xia, M.: Holographic algorithms with matchgates capture precisely tractable planar #CSP. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 427–436 (2010)

    Google Scholar 

  10. Cai, J.-Y., Lu, P., Xia, M.: Dichotomy for Holant* Problems with a Function on Domain Size 3. To appear in SODA 2013 (2013), Full paper available at http://arxiv.org/abs/1207.2354

  11. Dyer, M., Goldberg, L.A., Paterson, M.: On Counting Homomorphisms to Directed Acyclic Graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006, Part I. LNCS, vol. 4051, pp. 38–49. Springer, Heidelberg (2006); Journal of the ACM (JACM) 54(6), Article No. 27 (December 2007)

    Chapter  Google Scholar 

  12. Dyer, M.E., Greenhill, C.: The complexity of counting graph homomorphisms. Random Structures & Algorithms 17(3-4), 260–289 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dyer, M.E., Richerby, D.M.: On the Complexity of #CSP. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 725–734 (2010)

    Google Scholar 

  14. Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. SIAM Journal on Computing 39(7), 3336–3402 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, H., Williams, T.: The Complexity of Planar Boolean #CSP with Complex Weights, http://arxiv.org/abs/1212.2284

  16. Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press (2004)

    Google Scholar 

  17. Lovász, L.: Operations with structures. Acta Mathematica Hungarica 18, 321–328 (1967)

    Article  MATH  Google Scholar 

  18. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226 (1978)

    Google Scholar 

  19. Toda, S.: PP is as Hard as the Polynomial-Time Hierarchy. SIAM Journal on Computing 20, 865–877 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Valiant, L.G.: The Complexity of Computing the Permanent. Theoretical Computer Science (Elsevier) 8(2), 189–201

    Google Scholar 

  21. Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Valiant, L.G.: Accidental algorithms. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Wisconsin-Madison, WI, 53706, USA

    Jin-Yi Cai

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  1. Jin-Yi Cai
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Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Avinguda Catalunya, 35, 43002, Tarragona, Spain

    Adrian-Horia Dediu  & Carlos Martín-Vide  & 

  2. Fakultät für Informatik, Institut für Wissens- und Sprachverarbeitung, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Bianca Truthe

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Cai, JY. (2013). Complexity Dichotomy for Counting Problems. In: Dediu, AH., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2013. Lecture Notes in Computer Science, vol 7810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37064-9_1

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  • DOI: https://doi.org/10.1007/978-3-642-37064-9_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-37063-2

  • Online ISBN: 978-3-642-37064-9

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