, Volume 79, Issue 1, pp 96–138 | Cite as

On Kernelization and Approximation for the Vector Connectivity Problem

  • Stefan Kratsch
  • Manuel SorgeEmail author


In the Vector Connectivity problem we are given an undirected graph \(G=(V,E)\), a demand function \(\lambda :V\rightarrow \{0,\ldots ,d\}\), and an integer k. The question is whether there exists a set S of at most k vertices such that every vertex \(v\in V{\setminus } S\) has at least \(\lambda (v)\) vertex-disjoint paths to S; this abstractly captures questions about placing servers or warehouses relative to demands. The problem is \(\mathsf {NP}\)-hard already for instances with \(d=4\) (Cicalese et al., Theoretical Computer Science ’15), admits a log-factor approximation (Boros et al., Networks ’14), and is fixed-parameter tractable in terms of k (Lokshtanov, unpublished ’14). We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector d-Connectivity where the upper bound d on demands is a fixed constant. For Vector d-Connectivity we give a factor d-approximation algorithm and construct a vertex-linear kernelization, that is, an efficient reduction to an equivalent instance with \(f(d)k=O(k)\) vertices. For Vector Connectivity we have a factor \(\mathsf {opt} \)-approximation and we can show that it has no kernelization to size polynomial in k or even \(k+d\) unless \(\mathsf {NP} \subseteq \mathsf {coNP}/\mathsf {poly}\), which shows that \(f(d){\text {poly}}(k)\) is optimal for Vector d-Connectivity. Finally, we give a simple randomized fixed-parameter algorithm for Vector Connectivity with respect to k based on matroid intersection.


Parameterized complexity Approximation Graph algorithms Kernelization NP-hard problem Separators 



Funding was provided by Deutsche Forschungsgemeinschaft (DAPA (NI 369/12), PREMOD (KR 4286/1)).


  1. 1.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS ’09), pp. 629–638. IEEE Computer Society (2009)Google Scholar
  2. 2.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. CoRR, abs/0904.0727v3, (2013). URL
  3. 3.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28(1), 277–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boros, E., Heggernes, P., van ’t Hof, P., Milanic, M.: Vector connectivity in graphs. Networks 63(4), 277–285 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chitnis, R., Cygan, M., Hajiaghayi, M., Pilipczuk, M., Pilipczuk, M.: Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput. 45, 1171–1229 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cicalese, F., Milanic, M., Rizzi, R.: On the complexity of the vector connectivity problem. Theoret. Comput. Sci. 591, 60–71 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, 4th edn. Springer, Berlin (2010)Google Scholar
  9. 9.
    Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization lower bounds through colors and ids. ACM Trans. Algorithms 11(2), 13:1–13:20 (2014). doi: 10.1145/2650261 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Downey, R .G., Fellows, M .R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)zbMATHGoogle Scholar
  12. 12.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’10), pp. 503–510. SIAM (2010)Google Scholar
  13. 13.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs. In: Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science (STACS ’13, volume 20 of LIPIcs, pp. 92–103. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  14. 14.
    Hamacher, H .W., Drezner, Z.: Facility Location: Applications and Theory. Springer, Berlin (2002)zbMATHGoogle Scholar
  15. 15.
    Harant, J., Pruchnewski, A., Voigt, M.: On dominating sets and independent sets of graphs. Comb. Probab. Comput. 8, 547–553 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hermelin, D., Kratsch, S., Soltys, K., Wahlström, M., Wu, X.: A completeness theory for polynomial (Turing) kernelization. Algorithmica 71(3), 702–730 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jukna, S.: Extremal Combinatorics—With Applications in Computer Science. Texts in Theoretical Computer Science. Springer, Berlin (2001)zbMATHGoogle Scholar
  18. 18.
    Kim, E.J., Langer, A., Paul, C., Reidl, F., Rossmanith, P., Sau, I., Sikdar, S.: Linear kernels and single-exponential algorithms via protrusion decompositions. In: Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP ’13), volume 7965 of Lecture Notes in Computer Science, pp. 613–624. Springer (2013)Google Scholar
  19. 19.
    Kratsch, S., Sorge, M.: On kernelization and approximation for the vector connectivity problem. In: Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC’15), volume 43 of LIPIcs, pp. 377–388. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  20. 20.
    Kratsch, S., Wahlström, M.: Representative sets and irrelevant vertices: new tools for kernelization. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS ’12), pp. 450–459. IEEE Computer Society (2012)Google Scholar
  21. 21.
    Marx, D.: A parameterized view on matroid optimization problems. Theoret. Comput. Sci. 410(44), 4471–4479 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms 9(4), 30:1–30:35 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  24. 24.
    Oxley, J.: Matroid Theory. Oxford University Press, Oxford (2011)CrossRefzbMATHGoogle Scholar
  25. 25.
    Perfect, H.: Applications of Menger’s graph theorem. J. Math. Anal. Appl. 22(1), 96–111 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Skiena, S.: The Algorithm Design Manual, 2nd edn. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut für Informatik, Universität BonnBonnGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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