Abstract
The graph homomorphism problem (HOM) asks whether the vertices of a given n-vertex graph G can be mapped to the vertices of a given h-vertex graph H such that each edge of G is mapped to an edge of H. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 2-CSP problem. In this paper, we prove several lower bounds for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound \(2^{\Omega \left( \frac{n \log h}{\log \log h}\right) }\). This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound \(2^{\mathcal {O}(n\log {h})}\) is almost asymptotically tight.
We also investigate what properties of graphs G and H make it difficult to solve HOM(G, H). An easy observation is that an \(\mathcal {O}(h^n)\) upper bound can be improved to \(\mathcal {O}(h^{{\text {vc}}(G)})\) where \({\text {vc}}(G)\) is the minimum size of a vertex cover of G. The second lower bound \(h^{\Omega ({\text {vc}}(G))}\) shows that the upper bound is asymptotically tight. As to the properties of the “right-hand side” graph H, it is known that HOM(G, H) can be solved in time \((f(\Delta (H)))^n\) and \((f({\text {tw}}(H)))^n\) where \(\Delta (H)\) is the maximum degree of H and \({\text {tw}}(H)\) is the treewidth of H. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number \(\chi (H)\) does not exceed \({\text {tw}}(H)\) and \(\Delta (H)+1\), it is natural to ask whether similar upper bounds with respect to \(\chi (H)\) can be obtained. We provide a negative answer by establishing a lower bound \((f(\chi (H)))^n\) for every function f. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.
The full version of the paper is available at http://arxiv.org/abs/1502.05447
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References
Austrin, P.: Towards sharp inapproximability for any 2-CSP. SIAM J. Comput. 39(6), 2430–2463 (2010)
Barto, L., Kozik, M., Niven, T.: Graphs, polymorphisms and the complexity of homomorphism problems. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pp. 789–796 (2008)
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Computing 39(2), 546–563 (2009)
Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. J. Computer and System Sciences 72(8), 1346–1367 (2006)
Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoretical Computer Science 411(40–42), 3736–3756 (2010)
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer (2015)
Diaz, J., Serna, M., Thilikos, D.M.: Counting \(H\)-colorings of partial \(k\)-trees. Theoretical Computer Science 281, 291–309 (2002)
Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Computational complexity of the distance constrained labeling problem for trees (extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 294–305. Springer, Heidelberg (2008)
Fiala, J., Kratochvíl, J.: Locally constrained graph homomorphisms - structure, complexity, and applications. Computer Science Review 2(2), 97–111 (2008)
Fomin, F.V., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. Theory of Computing Systems 41(2), 381–393 (2007)
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Discrete Math. 5(4), 586–595 (1992)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)
Havet, F., Klazar, M., Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2, 1)-labeling of graphs. Algorithmica 59(2), 169–194 (2011)
Hell, P., Nešetřil, J.: On the complexity of \(H\)-coloring. J. Combinatorial Theory Ser. B 48(1), 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)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity. J. Computer and System Sciences 63(4), 512–530 (2001)
Junosza-Szaniawski, K., Kratochvíl, J., Liedloff, M., Rossmanith, P., Rzazewski, P.: Fast exact algorithm for l(2, 1)-labeling of graphs. Theor. Comput. Sci. 505, 42–54 (2013)
Lawler, E.L.: A note on the complexity of the chromatic number problem. Inf. Process. Lett. 5(3), 66–67 (1976)
Lokshtanov, D.: Private communication (2014)
Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 760–776. SIAM (2011)
Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of EATCS 3(105) (2013)
Lovász, L.: Large networks and graph limits, vol. 60. American Mathematical Soc. (2012)
Marx, D.: Can you beat treewidth? Theory of Computing 6(1), 85–112 (2010)
Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pp. 245–254 (2008)
Rzażewski, P.: Exact algorithm for graph homomorphism and locally injective graph homomorphism. Inf. Process. Lett. 114(7), 387–391 (2014)
Sipser, M.: Introduction to the Theory of Computation. Cengage Learning (2005)
Traxler, P.: The time complexity of constraint satisfaction. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 190–201. Springer, Heidelberg (2008)
Wahlström, M.: Problem 5.21. time complexity of graph homomorphism. In: Thore Husfeldt, Dieter Kratsch, R.P., Sorkin, G. (eds.) Exact Complexity of NP-Hard Problems. Dagstuhl Seminar 10441 Final Report. Dagstuhl (2010)
Wahlström, M.: New plain-exponential time classes for graph homomorphism. Theory of Computing Systems 49(2), 273–282 (2011)
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Fomin, F.V., Golovnev, A., Kulikov, A.S., Mihajlin, I. (2015). Lower Bounds for the Graph Homomorphism Problem. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_39
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