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Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes

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Abstract

We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #W[1]-hard on graphs G that are line graphs or bipartite graphs with degree 2 on one side. In our proofs, we use that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k length-2 paths into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes \(\mathcal {H}\) of pattern graphs, we complement this result: If the graphs in \(\mathcal {H}\) have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from \(\mathcal {H}\) is #W[1]-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

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Notes

  1. We need to assume here that the colors of the two identified edges agree.

  2. That is, r is bounded by dk for some overall constant d.

  3. Here, x is an indeterminate, so the quantity (7) is a polynomial in x.

References

  1. Cai, J.-Y., Zhiguo, F.: Holographic algorithm with matchgates is universal for planar #CSP over boolean domain. arXiv:abs/1603.07046 (2016)

  2. Cai, J.-Y., Hemachandra, L.A.: On the power of parity polynomial time. Mathematical Systems Theory 23(2), 95–106 (1990). https://doi.org/10.1007/BF02090768

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J.-Y., Lu, P.: Holographic algorithms: From art to science. J. Comput. Syst. Sci. 77(1), 41–61 (2011). https://doi.org/10.1016/j.jcss.2010.06.005

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, J.-Y., Lu, P., Xia, M.: Holographic algorithms by Fibonacci gates and holographic reductions for hardness. In: Proceedings of the 49th Annual Symposium on Foundations of Computer Science, FOCS, pp. 644–653. https://doi.org/10.1109/FOCS.2008.34 (2008)

  5. Cai, J.-Y., Pinyan, L., Xia, M.: Computational complexity of holant problems. SIAM J. Comput. 40(4), 1101–1132 (2011). https://doi.org/10.1137/100814585

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, J.-Y., Pinyan, L., Xia, M.: Holographic algorithms with matchgates capture precisely tractable planar #CSP. SIAM J. Comput. 46(3), 853–889 (2017). https://doi.org/10.1137/16M1073984

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D.W., Kanj, I.A., Ge, X.: Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput. 201(2), 216–231 (2005). https://doi.org/10.1016/j.ic.2005.05.001

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Thurley, M., Weyer, M.: Understanding the complexity of induced subgraph isomorphisms. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ICALP, pp. 587–596. https://doi.org/10.1007/978-3-540-70575-8_48 (2008)

  9. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006). https://doi.org/10.4007/annals.2006.164.51

    Article  MathSciNet  MATH  Google Scholar 

  10. Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Inf. Comput. 125(1), 1–12 (1996). https://doi.org/10.1006/inco.1996.0016

    Article  MathSciNet  MATH  Google Scholar 

  11. Curticapean, R.: Counting matchings of size k is #W[1]-hard. In: Proceedings of the 40th International Colloquium on Automata, Languages and Programming, ICALP, pp. 352–363. https://doi.org/10.1007/978-3-642-39206-1_30 (2013)

  12. Curticapean, R.: The Simple, Little and Slow Things Count: on Parameterized Counting Complexity. Saarland University, PhD thesis (2015)

    MATH  Google Scholar 

  13. Curticapean, R., Dell, H., Marx, D.: Homomorphisms are a good basis for counting small subgraphs. In: Proceedings of the 49th ACM Symposium on Theory of Computing, STOC, pp. 210–223. https://doi.org/10.1145/3055399.3055502 (2017)

  14. Curticapean, R., Marx, D.: Complexity of counting subgraphs Only the boundedness of the vertex-cover number counts. In: Proceedings of the 55th Annual Symposium on Foundations of Computer Science, FOCS, pp. 130–139, IEEE. https://doi.org/10.1109/FOCS.2014.22 (2014)

  15. Curticapean, R., Xia, M.: Parameterizing the permanent Genus, apices, minors, evaluation mod 2k. In: Proceedings of the 56th Annual Symposium on Foundations of Computer Science, FOCS, pp. 994–1009. https://doi.org/10.1109/FOCS.2015.65 (2015)

  16. Dagum, P., Luby, M.: Approximating the permanent of graphs with large factors. Theor. Comput. Sci. 102(2), 283–305 (1992). https://doi.org/10.1016/0304-3975(92)90234-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329(1-3), 315–323 (2004). https://doi.org/10.1016/j.tcs.2004.08.008

    Article  MathSciNet  MATH  Google Scholar 

  18. Dell, H., Husfeldt, T., Marx, D., Taslaman, N., Wahlen, M.: Exponential time complexity of the permanent and the Tutte polynomial. ACM Trans. Algorithms 10(4), 21 (2014). https://doi.org/10.1145/2635812

    Article  MathSciNet  MATH  Google Scholar 

  19. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004). https://doi.org/10.1137/S0097539703427203

    Article  MathSciNet  MATH  Google Scholar 

  20. Flum, J., Grohe, M.: Parameterized complexity theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  21. Frick, M.: Generalized model-checking over locally tree-decomposable classes. Theor. Comput. Sci. 37(1), 157–191 (2004). https://doi.org/10.1007/s00224-003-1111-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1), 1 (2007). https://doi.org/10.1145/1206035.1206036

    Article  MathSciNet  MATH  Google Scholar 

  23. Grohe, M., Schwentick, T., Segoufin, Luc: When is the evaluation of conjunctive queries tractable?. In: Proceedings of the 33rd ACM Symposium on Theory of Computing, STOC, pp. 657–666. https://doi.org/10.1145/380752.380867 (2001)

  24. Guruswami, V.: Maximum cut on line and total graphs. Discret. Appl. Math. 92(2-3), 217–221 (1999). https://doi.org/10.1016/S0166-218X(99)00056-6

    Article  MathSciNet  MATH  Google Scholar 

  25. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)

    Book  MATH  Google Scholar 

  26. Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. J. Stat. Phys. 48(1-2), 121–134 (1987). https://doi.org/10.1007/BF01010403

    Article  MathSciNet  Google Scholar 

  27. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4), 671–697 (2004). https://doi.org/10.1145/1008731.1008738

    Article  MathSciNet  MATH  Google Scholar 

  28. Kasteleyn, P.W. : Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, pp. 43–110. Academic Press (1967)

  29. Lehot, P.G.H.: An optimal algorithm to detect a line graph and output its root graph. J. ACM 21(4), 569–575 (1974). https://doi.org/10.1145/321850.321853

    Article  MathSciNet  MATH  Google Scholar 

  30. Lovász, L.: Large Networks and Graph Limits, volume 60 of Colloquium Publications. American Mathematical Society. http://www.ams.org/bookstore-getitem/item=COLL-60 (2012)

  31. Lozin, V.V., Mosca, R.: Independent sets in extensions of 2K 2-free graphs. Discret. Appl. Math. 146(1), 74–80 (2005). https://doi.org/10.1016/j.dam.2004.07.006

    Article  MATH  Google Scholar 

  32. Meeks, K.: The challenges of unbounded treewidth in parameterised subgraph counting problems. Discret. Appl. Math. 198, 170–194 (2016). https://doi.org/10.1016/j.dam.2015.06.019

    Article  MathSciNet  MATH  Google Scholar 

  33. Roth, M.: Counting restricted homomorphisms via Möbius inversion over matroid lattices. In: 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pp. 63:1–63:14. https://doi.org/10.4230/LIPIcs.ESA.2017.63 (2017)

  34. Sbihi, N.: Algorithme de recherche d’un stable de cardinalite maximum dans un graphe sans etoile. Discret. Math. 29, 53–76 (1980). https://doi.org/10.1016/0012-365X(90)90287-R

    Article  MATH  Google Scholar 

  35. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics - an exact result. Philos. Mag. 6(68), 1478–6435 (1961). https://doi.org/10.1080/14786436108243366

    Article  MathSciNet  MATH  Google Scholar 

  36. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001). https://doi.org/10.1137/S0097539797321602

    Article  MathSciNet  MATH  Google Scholar 

  37. Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979). https://doi.org/10.1016/0304-3975(79)90044-6

    Article  MathSciNet  MATH  Google Scholar 

  38. Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008). https://doi.org/10.1137/070682575

    Article  MathSciNet  MATH  Google Scholar 

  39. Šoltés, Ľ.: Forbidden induced subgraphs for line graphs. Discret. Math. 132(1), 391–394 (1994). https://doi.org/10.1016/0012-365X(92)00577-E

    Article  MathSciNet  MATH  Google Scholar 

  40. Williams, V.V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. SIAM J. Comput. 42(3), 831–854 (2013). https://doi.org/10.1137/09076619X

    Article  MathSciNet  MATH  Google Scholar 

  41. Xia, M., Zhang, P., Zhao, W.: Computational complexity of counting problems on 3-regular planar graphs. Theor. Comput. Sci. 384(1), 111–125 (2007). Theory and Applications of Models of Computation. https://doi.org/10.1016/j.tcs.2007.05.023

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, X.-D., Bylka, S.: Disjoint triangles of a cubic line graph. Graphs and Combinatorics 20(2), 275–280 (2004). https://doi.org/10.1007/s00373-004-0551-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors thank Cornelius Brand and Markus Bläser for interesting discussions, and Johannes Schmitt for pointing out a proof of Lemma 20 and allowing us to use it in this paper.

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Correspondence to Radu Curticapean.

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This article is part of the Topical Collection on Special Issue on Theoretical Aspects of Computer Science (STACS 2017)

Most of this work was done while the authors were visiting the Simons Institute for the Theory of Computing. Radu Curticapean is supported by ERC grants PARAMTIGHT (No. 280152) and SYSTEMATICGRAPH (No. 725978) and VILLUM Foundation grant 16582.

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Curticapean, R., Dell, H. & Roth, M. Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes. Theory Comput Syst 63, 987–1026 (2019). https://doi.org/10.1007/s00224-018-9893-y

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