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Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

Published:26 June 2015Publication History
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Abstract

We consider the problem of discovering overlapping communities in networks that we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems, we seek a collection SS consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U belongs to at most t sets of S, while in r-Set Packing with t-Overlap, each pair of sets in S overlaps in at most t elements. For both problems, each set of S has at most r elements.

Similarly, both of our Graph Packing problems seek a collection K of at least k subgraphs in a graph G, each isomorphic to a graph HH. In H-Packing with t-Membership, each vertex of G belongs to at most t subgraphs of K, while in H-Packing with t-Overlap, each pair of subgraphs in K overlaps in at most t vertices. For both problems, each member of H has at most r vertices and m edges, where t, r, and m are constants.

Here, we show NP-completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the H-Packing with t-Membership problem analogous to the Kirkpatrick and Hell dichotomy [Kirkpatrick and Hell 1978]. Using polynomial parameter transformations, we reduce the r-Set Packing with t-Membership to a problem kernel with O((r + 1)rkr) elements and the H-Packing with t-Membership and its edge version to problem kernels with O((r + 1)rkr) and O((m + 1)mkm) vertices, respectively. On the other hand, by generalizing [Fellows et al. 2008; Moser 2009], we achieve a kernel with O(rrkrt − 1) elements for the r-Set Packing with t-Overlap and kernels with O(rrkrt − 1) and O(mmkmt − 1) vertices for the H-Packing with t-Overlap and its edge version, respectively. In all cases, k is the input parameter, while t, r, and m are constants.

References

  1. F. N. Abu-Khzam. 2010. An improved kernelization algorithm for r-Set Packing. Inform. Process. Lett. 110, 16 (2010), 621--624. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. S. Banerjee and S. Khuller. 2001. A clustering scheme for hierarchical control in multi-hop wireless networks. In Proceedings of the 20th Joint Conference of the IEEE Computer and Communications Societies (INFOCOM’01), Vol. 2. IEEE Society Press, 1028--1037.Google ScholarGoogle Scholar
  3. H. L. Bodlaender, S. Thomassé, and A. Yeo. 2008. Analysis of Data Reduction: Transformations Give Evidence for Non-existence of Polynomial Kernels. Technical Report UU-CS-2008-030. Department of Information and Computer Sciences, Utrecht University.Google ScholarGoogle Scholar
  4. A. Caprara and R. Rizzi. 2002. Packing triangles in bounded degree graphs. Inform. Process. Lett. 84, 4 (2002), 175--180. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. J. Chen, H. Fernau, P. Shaw, J. Wang, and Z. Yang. 2012. Kernels for packing and covering problems. In Frontiers in Algorithmics and Algorithmic Aspects in Information and Management (LNCS), J. Snoeyink, P. Lu, K. Su, and L. Wang (Eds.), Vol. 7825. Springer, Berlin, 199--211. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. H. Dell and D. Marx. 2012. Kernelization of packing problems. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12), Y. Rabani (Ed.). SIAM, 68--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. D. Dor and M. Tarsi. 1997. Graph decomposition is NP-complete: A complete proof of Holyer’s conjecture. SIAM J. Comput. 26, 4 (1997), 1166--1187. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. R. G. Downey, J. Egan, M. R. Fellows, F. A. Rosamond, and P. Shaw. 2014. Dynamic dominating set and turbo-charging greedy heuristics. Tsinghua Sci. Technol. 19, 4 (2014), 329--337.Google ScholarGoogle Scholar
  9. M. Fellows, J. Guo, C. Komusiewicz, R. Niedermeier, and J. Uhlmann. 2011. Graph-based data clustering with overlaps. Discrete Optimization 8, 1 (2011), 2--17. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. M. Fellows, P. Heggernes, F. Rosamond, C. Sloper, and J. A. Telle. 2004. Finding k disjoint triangles in an arbitrary graph. In the 30th Workshop on Graph-Theoretic Concepts in Computer Science (LNCS), J. Hromkovič, M. Nagl, and B. Westfechtel (Eds.), Vol. 3353. Springer, Berlin, 235--244. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. M. Fellows, C. Knauer, N. Nishimura, P. Ragde, F. Rosamond, U. Stege, D. Thilikos, and S. Whitesides. 2008. Faster fixed-parameter tractable algorithms for matching and packing problems. Algorithmica 52, 2 (2008), 167--176. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. H. Fernau and D. Raible. 2009. A parameterized perspective on packing paths of length two. J. Combinatorial Optimization 18, 4 (2009), 319--341.Google ScholarGoogle ScholarCross RefCross Ref
  13. A. C. Giannopoulou, B. M. P. Jansen, D. Lokshtanov, and S. Saurabh. 2014. Uniform Kernelization Complexity of Hitting Forbidding Minors. (2014). Unpublished, see http://www.win.tue.nl/∼bjansen/publications.html.Google ScholarGoogle Scholar
  14. D. Hermelin and Xi Wu. 2012. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12), Y. Rabani (Ed.). SIAM, 104--113. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. I. Holyer. 1981. The NP-completeness of some edge-partition problems. SIAM J. Comput. 10, 4 (1981), 713--717.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. B. M. P. Jansen and D. Marx. 2014. Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels. CoRR abs/1410.0855 (2014). Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. D. G. Kirkpatrick and P. Hell. 1978. On the completeness of a generalized matching problem. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC’78). ACM, 240--245. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. S. Micali and V. V. Vazirani. 1980. An O(&surd;|V||E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science (SFCS’80). IEEE Computer Society, 17--27. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. H. Moser. 2009. A problem kernelization for graph packing. In the 35th International Conference on Current Trends in Theory and Practice of Computer Science (LNCS), M. Nielsen, A. Kučera, P. B. Miltersen, C. Palamidessi, P. Tůma, and F. Valencia (Eds.), Vol. 5404. Springer, Berlin, 401--412. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. G. Palla, I. Derényi, I. Farkas, and T. Vicsek. 2005. Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 7043 (2005), 814--818.Google ScholarGoogle ScholarCross RefCross Ref
  21. E. Prieto and C. Sloper. 2006. Looking at the stars. Theor. Comput. Sci. 351, 3 (2006), 437--445. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. J. Romero and A. López-Ortiz. 2014a. The G-packing with t-overlap problem. In the 8th International Workshop on Algorithms and Computation (LNCS), S. P. Pal and K. Sadakane (Eds.), Vol. 8344. Springer, Berlin, 114--124.Google ScholarGoogle Scholar
  23. J. Romero and A. López-Ortiz. 2014b. A parameterized algorithm for packing overlapping subgraphs. In the 9th International Computer Science Symposium in Russia (LNCS), E. A. Hirsch, S. O. Kuznetsov, J.-É. Pin, and N. K. Vereshchagin (Eds.), Vol. 8476. Springer, Berlin, 325--336.Google ScholarGoogle Scholar
  24. Y. Shiloach. 1981. Another look at the degree constrained subgraph problem. Inform. Process. Lett. 12, 2 (1981), 89--92.Google ScholarGoogle ScholarCross RefCross Ref
  25. J. Wang, D. Ning, Q. Feng, and J. Chen. 2010. An improved kernelization for P2-packing. Inform. Process. Lett. 110, 5 (2010), 188--192.Google ScholarGoogle ScholarCross RefCross Ref

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