Parity Separation: A Scientifically Proven Method for Permanent Weight Loss

Author Radu Curticapean

Thumbnail PDF


  • Filesize: 0.59 MB
  • 14 pages

Document Identifiers

Author Details

Radu Curticapean

Cite AsGet BibTex

Radu Curticapean. Parity Separation: A Scientifically Proven Method for Permanent Weight Loss. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and -1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. Our technique is based upon the Holant framework and matchgates. We derive the following applications: Firstly, an alternative #P-completeness proof for counting unweighted perfect matchings. Secondly, C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the “equality-testing version” of any natural counting problem that is not already #P-hard under parsimonious reductions. Thirdly, an alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH.
  • perfect matchings
  • counting complexity
  • structural complexity
  • exponentialtime hypothesis


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


  1. Amir Ben-Dor and Shai Halevi. Zero-one permanent is #P-complete, A simpler proof. In Second Israel Symposium on Theory of Computing Systems, ISTCS 1993, Natanya, Israel, June 7-9, 1993. Proceedings, pages 108-117, 1993. Google Scholar
  2. Markus Bläser and Radu Curticapean. The complexity of the cover polynomials for planar graphs of bounded degree. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, pages 96-107, 2011. URL:
  3. Markus Bläser and Radu Curticapean. Weighted counting of k-matchings is #W[1]-hard. In Parameterized and Exact Computation - 7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceedings, pages 171-181, 2012. URL:
  4. Markus Bläser and Holger Dell. Complexity of the cover polynomial. In Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings, pages 801-812, 2007. URL:
  5. Graham Brightwell and Peter Winkler. Counting linear extensions is #P-complete. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 175-181, 1991. URL:
  6. Jin-Yi Cai and Pinyan Lu. Holographic algorithms: From art to science. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 401-410, New York, NY, USA, 2007. ACM. URL:
  7. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holographic algorithms by Fibonacci gates and holographic reductions for hardness. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 644-653. IEEE Computer Society, 2008. URL:
  8. Mike Chen. The complexity of checking whether two DAG have the same number of topological sorts, November 2010. URL:
  9. Radu Curticapean. Block interpolation: A framework for tight exponential-time counting complexity. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 380-392, 2015. URL:
  10. Radu Curticapean. Parity separation: A scientifically proven method for permanent weight loss. CoRR, abs/1511.07480, 2015. URL:
  11. Radu Curticapean. The simple, little and slow things count: on parameterized counting complexity. PhD thesis, Saarland University, August 2015. Google Scholar
  12. Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1650-1669, 2016. URL:
  13. Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential time complexity of the permanent and the Tutte polynomial. ACM Transactions on Algorithms, 10(4):21, 2014. URL:
  14. Jack Edmonds. Paths, trees, and flowers. In Classic Papers in Combinatorics, Modern Birkhauser Classics, pages 361-379. Birkhauser Boston, 1987. URL:
  15. Lane A. Hemaspaandra and Mitsunori Ogihara. The Complexity Theory Companion. Springer, 2002. Google Scholar
  16. Lane A. Hemaspaandra and Heribert Vollmer. The satanic notations: counting classes beyond #P and other definitional adventures. SIGACT News, 26(1):2-13, 1995. URL:
  17. Christian Hoffmann. Exponential time complexity of weighted counting of independent sets. In Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, pages 180-191, 2010. URL:
  18. Thore Husfeldt and Nina Taslaman. The exponential time complexity of computing the probability that a graph is connected. In Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, pages 192-203, 2010. URL:
  19. Russel Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL:
  20. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL:
  21. Pieter W. Kasteleyn. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica, 27(12):1209-1225, 1961. URL:
  22. Pieter W. Kasteleyn. Graph Theory and Crystal Physics. In Graph Theory and Theoretical Physics, pages 43-110. Academic Press, 1967. Google Scholar
  23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011. URL:
  24. Patrick Scharpfenecker and Jacobo Torán. Solution-graphs of boolean formulas and isomorphism. Electronic Colloquium on Computational Complexity (ECCC), 23:24, 2016. URL:
  25. J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell University, 1975. Google Scholar
  26. H. N. V. Temperley and Michael E. Fisher. Dimer problem in statistical mechanics - an exact result. Philosophical Magazine, 6(68):1478-6435, 1961. Google Scholar
  27. Leslie G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979. Google Scholar
  28. Leslie G. Valiant. Accidental algorithms. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 509-517, 2006. URL:
  29. Leslie G. Valiant. Holographic algorithms. SIAM Journal on Computing, 37(5):1565-1594, 2008. URL:
  30. Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986. URL:
  31. Klaus W. Wagner. The complexity of combinatorial problems with succinct input representation. Acta Informatica, 23(3):325-356, 1986. URL:
  32. Viktoria Zanko. #P-completeness via many-one reductions. International Journal of Foundations of Computer Science, 2(1):77-82, 1991. URL: