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On Tree-Constrained Matchings and Generalizations

  • Stefan Canzar
  • Khaled Elbassioni
  • Gunnar W. Klau
  • Julián Mestre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We consider the following Tree-Constrained Bipartite Matching problem: Given two rooted trees T 1 = (V 1,E 1), T 2 = (V 2,E 2) and a weight function w: V 1×V 2 ↦ℝ + , find a maximum weight matching \(\mathcal{M}\) between nodes of the two trees, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is \(\mathcal{APX}\)-hard and thus, unless \(\mathcal{P} = \mathcal{NP}\), disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a 2-approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of 2 − o(1). In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a 2-approximation for the k-dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by ρ. We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on ρ is most likely unavoidable.

Keywords

Interval Graph Basic Feasible Solution Bipartite Match Matched Node Local Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. Journal of Computer and System Sciences 54(2), 317–331 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. Journal of the ACM 48, 1069–1090 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM Journal on Computing 36(1), 1–15 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berman, P., Karpinski, M.: On some tighter inapproximability results. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Canzar, S., Elbassioni, K., Klau, G.W., Mestre, J.: On tree-constrained matchings and generalizations. Technical Report MAC1102, Centrum Wiskunde & Informatica (CWI), Amsterdam, the Netherlands (2011)Google Scholar
  6. 6.
    Feige, U., Lovász, L.: Two-prover one-round proof systems: their power and their problems. In: Proc. of STOC, pp. 733–744. ACM, New York (1992)Google Scholar
  7. 7.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combina- torial Optimization. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-dimensional matching. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 83–97. Springer, Heidelberg (2003)Google Scholar
  9. 9.
    Kortsarz, G.: On the hardness of approximating spanners. Algorithmica 30(3), 432–450 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mosig, A., Jäger, S., Wang, C., Nath, S., Ersoy, I., Palaniappan, K., Chen, S.-S.: Tracking cells in life cell imaging videos using topological alignments. Algorithms for Molecular Biology 4, 10 (2009)CrossRefGoogle Scholar
  11. 11.
    Wolsey, L.A., Nemhauser, G.L.: Integer and Combinatorial Optimization. Wiley Interscience, Hoboken (1999)zbMATHGoogle Scholar
  12. 12.
    Xiao, H., Li, Y., Du, J., Mosig, A.: Ct3d: Tracking Microglia Motility in 3D Using a Novel Cosegmentation Approach. Bioinformatics 27(4), 564–571 (2011)CrossRefGoogle Scholar
  13. 13.
    Yang, F., Mackey, M.A., Ianzini, F., Gallardo, G., Sonka, M.: Cell segmentation, tracking, and mitosis detection using temporal context. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 302–309. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Zimmer, C., Zhang, B., Dufour, A., Thebaud, A., Berlemont, S., Meas-Yedid, V., Marin, J.-C.O.: On the digital trail of mobile cells. Signal Processing Magazine 23(3), 54–62 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Stefan Canzar
    • 1
  • Khaled Elbassioni
    • 2
  • Gunnar W. Klau
    • 1
  • Julián Mestre
    • 3
  1. 1.Centrum Wiskunde & InformaticaLife Sciences GroupAmsterdamThe Netherlands
  2. 2.Algorithms and Complexity Dept.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.School of Information TechnologiesThe University of SydneyAustralia

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