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A Slice Theoretic Approach for Embedding Problems on Digraphs

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9224)

Abstract

We say that a digraph H can be covered by k paths if there exist k directed paths \(\mathfrak {p}_1,\mathfrak {p}_2,\ldots ,\mathfrak {p}_k\) such that \(H=\cup _{i=1}^k \mathfrak {p}_i\). In this work we devise parameterized algorithms for embedding problems on digraphs in the setting in which the host digraph G has directed pathwidth w and the pattern digraph H can be covered by k paths. More precisely, we show that the subgraph isomorphism, subgraph homeomorphism, and two other related embedding problems can each be solved in time \(2^{O(k\cdot w \log k\cdot w)} \cdot |H|^{O(k\cdot w)}\cdot |G|^{O(k\cdot w)}\). We note in particular that for constant values of w and k, our algorithm runs in polynomial time with respect to the size of the pattern digraph H. Therefore for the classes of digraphs considered in this work our results yield an exponential speedup with respect to the best general algorithm for the subgraph isomorphism problem which runs in time \(O^*(2^{|H|}\cdot |G|^{ tw (H)})\) (where \( tw (H)\) is the undirected treewidth of H), and an exponential speedup with respect to the best general algorithm for the subgraph homeomorphism problem which runs in time \(|G|^{O(|H|)}\).

Keywords

  • Directed pathwidth
  • Subgraph isomorphism
  • Subgraph homeomorphism
  • Slice languages

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References

  1. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM (JACM) 42(4), 844–856 (1995)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics 22(2), 161–172 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: Handbook of Theoretical Computer Science, Chap. 5, pp. 194–242. Elsevier, Amsterdam (1990)

    Google Scholar 

  5. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theor. Comput. Syst. 33(2), 125–150 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. de Oliveira Oliveira, M.: Hasse diagram generators and Petri nets. Fundamenta Informaticae 105(3), 263–289 (2010)

    MathSciNet  MATH  Google Scholar 

  7. de Oliveira Oliveira, M.: Canonizable partial order generators. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 445–457. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  8. de Oliveira Oliveira, M.: Subgraphs satisfying MSO properties on z-topologically orderable digraphs. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 123–136. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  9. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S., Rao, B.V.R.: Faster algorithms for finding and counting subgraphs. J. Comput. Syst. Sci. 78(3), 698–706 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Ganian, R., Hliněný, P., Kneis, J., Langer, A., Obdržálek, J., Rossmanith, P.: On digraph width measures in parameterized algorithmics. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 185–197. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  12. Grohe, M., Kawarabayashi, K.-i., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: STOC, pp. 479–488. ACM (2011)

    Google Scholar 

  13. Gruber, H.: Digraph complexity measures and applications in formal language theory. Discrete Math. Theor. Comput. Sci. 14(2), 189–204 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Gupta, A., Nishimura, N., Proskurowski, A., Ragde, P.: Embeddings of k-connected graphs of pathwidth k. Discrete Appl. Math. 145(2), 242–265 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theor. Ser. B 82(1), 138–154 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Tamaki, H.: A polynomial time algorithm for bounded directed pathwidth. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 331–342. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  17. Vardi, M.Y.: The complexity of relational query languages. In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pp. 137–146. ACM (1982)

    Google Scholar 

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Acknowledgements

I gratefully acknowledge financial support from the European Research Council, ERC grant agreement 339691, within the context of the project Feasibility, Logic and Randomness (FEALORA).

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Correspondence to Mateus de Oliveira Oliveira .

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de Oliveira Oliveira, M. (2016). A Slice Theoretic Approach for Embedding Problems on Digraphs. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_26

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  • DOI: https://doi.org/10.1007/978-3-662-53174-7_26

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