Effective and Efficient Data Reduction for the Subset Interconnection Design Problem

  • Jiehua Chen
  • Christian Komusiewicz
  • Rolf Niedermeier
  • Manuel Sorge
  • Ondřej Suchý
  • Mathias Weller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


The NP-hard Subset Interconnection Design problem is motivated by applications in designing vacuum systems and scalable overlay networks. It has as input a set V and a collection of subsets V1, V2, …, Vm, and asks for a minimum-cardinality edge set E such that for the graph G = (V,E) all induced subgraphs G[V1], G[V2], …, G[Vm] are connected. It has also been studied under the name Minimum Topic-Connected Overlay. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve optimality. Our contribution is threefold: First, we point out flaws in earlier polynomial-time data reduction rules. Second, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. Third, we show linear-time solvability in case of a constant number m of subsets, implying fixed-parameter tractability for the parameter m. To achieve our results, we elaborate on polynomial-time data reduction rules (partly “repairing” previous flawed ones) which also may be of practical use in solving Subset Interconnection Design.


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  1. 1.
    Angluin, D., Aspnes, J., Reyzin, L.: Inferring social networks from outbreaks. In: Hutter, M., Stephan, F., Vovk, V., Zeugmann, T. (eds.) ALT 2010. LNCS (LNAI), vol. 6331, pp. 104–118. Springer, Heidelberg (2010)Google Scholar
  2. 2.
    Chockler, G., Melamed, R., Tock, Y., Vitenberg, R.: Constructing scalable overlays for pub-sub with many topics. In: Proc. 26th PODC, pp. 109–118. ACM (2007)Google Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  4. 4.
    Du, D.-Z.: An optimization problem on graphs. Discrete Appl. Math. 14(1), 101–104 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Du, D.-Z., Kelley, D.F.: On complexity of subset interconnection designs. J. Global Optim. 6(2), 193–205 (1995)MathSciNetMATHGoogle Scholar
  6. 6.
    Du, D.-Z., Miller, Z.: Matroids and subset interconnection design. SIAM J. Discrete Math. 1(4), 416–424 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fan, H., Wu, Y.-L.: Interconnection graph problem. In: Proc. FCS 2008, pp. 51–55. CSREA Press (2008)Google Scholar
  8. 8.
    Fan, H., Hundt, C., Wu, Y.-L., Ernst, J.: Algorithms and implementation for interconnection graph problem. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 201–210. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  10. 10.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  11. 11.
    Hosoda, J., Hromkovič, J., Izumi, T., Ono, H., Steinová, M., Wada, K.: On the approximability and hardness of minimum topic connected overlay and its special instances. Theor. Comput. Sci. 429, 144–154 (2012)CrossRefMATHGoogle Scholar
  12. 12.
    Korach, E., Stern, M.: The clustering matroid and the optimal clustering tree. Math. Program. 98(1-3), 385–414 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  14. 14.
    Onus, M., Richa, A.W.: Minimum maximum-degree publish-subscribe overlay network design. IEEE/ACM Trans. Netw. 19(5), 1331–1343 (2011)CrossRefGoogle Scholar
  15. 15.
    Tang, T.-Z.: An optimality condition for minimum feasible graphs. Applied Mathematics - A Journal of Chinese Universities, 24–21 (1989) (in Chinese)Google Scholar
  16. 16.
    Xu, Y., Fu, X.: On the minimum feasible graph for four sets. Applied Mathematics - A Journal of Chinese Universities 10, 457–462 (1995)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jiehua Chen
    • 1
  • Christian Komusiewicz
    • 1
  • Rolf Niedermeier
    • 1
  • Manuel Sorge
    • 1
  • Ondřej Suchý
    • 2
  • Mathias Weller
    • 3
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany
  2. 2.Department of Theoretical Computer ScienceCzech Technical University in PragueCzech Republic
  3. 3.Département InformatiqueLIRMMFrance

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