The Complexity of Homomorphism Indistinguishability

Authors Jan Böker , Yijia Chen , Martin Grohe , Gaurav Rattan



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2019.54.pdf
  • Filesize: 0.51 MB
  • 13 pages

Document Identifiers

Author Details

Jan Böker
  • RWTH Aachen University, Aachen, Germany
Yijia Chen
  • Fudan University, Shanghai, China
Martin Grohe
  • RWTH Aachen University, Aachen, Germany
Gaurav Rattan
  • RWTH Aachen University, Aachen, Germany

Cite AsGet BibTex

Jan Böker, Yijia Chen, Martin Grohe, and Gaurav Rattan. The Complexity of Homomorphism Indistinguishability. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.54

Abstract

For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018]. In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable. Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph theory
Keywords
  • graph homomorphism numbers
  • counting complexity
  • treewidth

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. V. Arvind, J. Köbler, S. Kuhnert, and Y. Vasudev. Approximate Graph Isomorphism. In B. Rovan, V. Sassone, and P. Widmayer, editors, Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, volume 7464 of Lecture Notes in Computer Science, pages 100-111. Springer Verlag, 2012. Google Scholar
  2. L. Babai. Graph Isomorphism in Quasipolynomial Time. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC '16), pages 684-697, 2016. Google Scholar
  3. J. Böker. Color Refinement, Homomorphisms, and Hypergraphs. arXiv e-prints, page arXiv:1903.12432, March 2019. Google Scholar
  4. R. Curticapean. Parity Separation: A Scientifically Proven Method for Permanent Weight Loss. In I. Chatzigiannakis, M. Mitzenmacher, Y. Rabani, and D. Sangiorgi, editors, Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (Track A), volume 55 of LIPIcs, pages 47:1-47:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  5. V. Dalmau and P. Jonsson. The complexity of counting homomorphisms seen from the other side. Theoretical Computer Science, 329(1-3):315-323, 2004. Google Scholar
  6. H. Dell, M. Grohe, and G. Rattan. Lovász Meets Weisfeiler and Leman. In I. Chatzigiannakis, C. Kaklamanis, D. Marx, and D. Sannella, editors, Proceedings of the 45th International Colloquium on Automata, Languages and Programming (Track A), volume 107 of LIPIcs, pages 40:1-40:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  7. R. Diestel. Graph Theory. Springer Verlag, 4th edition, 2010. Google Scholar
  8. M. Grohe, K. Kersting, M. Mladenov, and P. Schweitzer. Color Refinement and its Applications. In G. Van den Broeck, K. Kersting, S. Natarajan, and D. Poole, editors, An Introduction to Lifted Probabilistic Inference. Cambridge University Press, 2017. To appear. URL: https://lii.rwth-aachen.de/images/Mitarbeiter/pub/grohe/cr.pdf.
  9. M. Grohe, G. Rattan, and G. Woeginger. Graph Similarity and Approximate Isomorphism. In I. Potapov, P.G. Spirakis, and J. Worrell, editors, Proceedings of the 43rd International Symposium on Mathematical Foundations of Computer Science, volume 117 of LIPIcs, pages 20:1-20:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  10. L. A. Hemaspaandra and H. Vollmer. The satanic notations: counting classes beyond #P and other definitional adventures. SIGACT News, 26(1):2-13, 1995. Google Scholar
  11. A. Kolla, I. Koutis, V. Madan, and A.K. Sinop. Spectrally Robust Graph Isomorphism. In I. Chatzigiannakis, C. Kaklamanis, D. Marx, and D. Sannella, editors, Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, volume 107 of LIPIcs, pages 84:1-84:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  12. L. Lovász. Operations with Structures. Acta Mathematica Hungarica, 18:321-328, 1967. Google Scholar
  13. L. Lovász. Large Networks and Graph Limits. American Mathematical Society, 2012. Google Scholar
  14. V. Nagarajan and M. Sviridenko. On the maximum quadratic assignment problem. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 516-524, 2009. Google Scholar
  15. R. O'Donnell, J. Wright, C. Wu, and Y. Zhou. Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1659-1677, 2014. Google Scholar
  16. N. Shervashidze, P. Schweitzer, E.J. van Leeuwen, K. Mehlhorn, and K.M. Borgwardt. Weisfeiler-Lehman Graph Kernels. Journal of Machine Learning Research, 12:2539-2561, 2011. Google Scholar
  17. N. Shervashidze, S. Vishwanathan, T. Petri, K. Mehlhorn, and K. Borgwardt. Efficient graphlet kernels for large graph comparison. In Artificial Intelligence and Statistics, pages 488-495, 2009. Google Scholar
  18. J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell University, 1975. Google Scholar
  19. S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991. Google Scholar
  20. E. R. Van Dam and W. H. Haemers. Which graphs are determined by their spectrum? Linear Algebra and its applications, 373:241-272, 2003. URL: https://doi.org/10.1016/S0024-3795(03)00483-X.
  21. S.V.N. Vishwanathan, N.N. Schraudolph, R. Kondor, and K.M. Borgwardt;. Graph Kernels. Journal of Machine Learning Research, 11:1201-1242, 2010. Google Scholar
  22. K. W. Wagner. Some Observations on the Connection Between Counting an Recursion. Theoretical Computuer Science, 47(3):131-147, 1986. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail