Abstract
Counting homomorphisms between graphs has applications in variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this work we show a new application of counting graph homomorphisms to the areas of exact and parameterized algorithms.
We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser’s formula for counting perfect matchings of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42, 844–856 (1995)
Babai, L.: Moderately exponential bound for graph isomorphism. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 34–50. Springer, Heidelberg (1981)
Babai, L., Kantor, W.M., Luks, E.M.: Computational complexity and the classification of finite simple groups. In: FOCS, pp. 162–171 (1983)
Bax, E.T.: Algorithms to count paths and cycles. Inform. Process. Lett. 52, 249–252 (1994)
Beals, R., Chang, R., Gasarch, W.I., Torán, J.: On finding the number of graph automorphisms. Chicago J. Theor. Comput. Sci. (1999)
Bernardi, O., Noy, M., Welsh, D.: On the growth rate of minor-closed classes of graphs, http://arxiv.org/abs/0710.2995
Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)
Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: FOCS, pp. 575–582 (2006)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution. In: STOC, pp. 67–74 (2007)
Blankenship, R., Oporowski, B.: Book embeddings of graphs and minor-closed classes. In: Proceedings of the 32nd Southeastern International Conference on Combinatorics, Graph Theory and Computing
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Counting graph homomorphisms. In: Topics in discrete mathematics. Algorithms Combin., vol. 26, pp. 315–371. Springer, Berlin (2006)
Cygan, M., Pilipczuk, M.: Faster exact bandwidth. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 101–109. Springer, Heidelberg (2008)
Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theoret. Comput. Sci. 329, 315–323 (2004)
Díaz, J., Serna, M.J., Thilikos, D.M.: Counting h-colorings of partial k-trees. Theor. Comput. Sci. 281, 291–309 (2002)
Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. Random Structures Algorithms 17, 260–289 (2000)
Feige, U.: Coping with the NP-Hardness of the Graph Bandwidth Problem. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 10–19. Springer, Heidelberg (2000)
Feige, U., Hajiaghayi, M.T., Lee, J.R.: Improved approximation algorithms for minimum-weight vertex separators. In: STOC, pp. 563–572 (2005)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54, Art. 1, 24 (electronic) (2007)
Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Combin. Theory Ser. B 48, 92–110 (1990)
Hell, P., Nešetřil, J.: Graphs and homomorphisms. Oxford Lecture Series in Mathematics and its Applications, vol. 28. Oxford University Press, Oxford (2004)
Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett. 1, 49–51 (1982)
Kohn, S., Gottlieb, A., Kohn, M.: A generating function approach to the traveling salesman problem. In: Proceedings of the annual ACM conference, pp. 294–300 (1977)
Koivisto, M.: An O(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: FOCS, pp. 583–590 (2006)
Lovász, L.: Operations with structures. Acta Math. Hung. 18, 321–328 (1967)
Malitz, S.M.: Genus g graphs have pagenumber \(O(\sqrt{g})\). J. Algorithms 17, 85–109 (1994)
Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: FOCS, pp. 182–191 (1995)
Norine, S., Seymour, P.D., Thomas, R., Wollan, P.: Proper minor-closed families are small. J. Comb. Theory, Ser. B 96, 754–757 (2006)
Ryser, H.J.: Combinatorial mathematics. The Carus Mathematical Monographs, vol. 14. The Mathematical Association of America (1963)
Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189–201 (1979)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Amini, O., Fomin, F.V., Saurabh, S. (2009). Counting Subgraphs via Homomorphisms. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-02927-1_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02926-4
Online ISBN: 978-3-642-02927-1
eBook Packages: Computer ScienceComputer Science (R0)