On Tree-Constrained Matchings and Generalizations

Abstract

We consider the following Tree-Constrained Bipartite Matching problem: Given a bipartite graph G=(V 1,V 2,E) with edge weights \(w:E \mapsto\mathbb{R}_{+}\), a rooted tree T 1 on the set V 1 and a rooted tree T 2 on the set V 1, find a maximum weight matching \(\mathcal{M}\) in G, 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.

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Notes

  1. 1.

    In fact, their algorithm is slightly different. First, it does not compute an optimal solution in each iteration; second, it only removes edges with negative weight. Our improved approximation guarantees require that we re-compute the optimal basic feasible solution in each iteration.

References

  1. 1.

    Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54(2), 317–331 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36(1), 1–15 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Berman, P., Karpinski, M.: On some tighter inapproximability results. In: Proc. of the 26th International Colloquium on Automata, Languages and Programming, pp. 200–209 (1999)

    Google Scholar 

  4. 4.

    Canzar, S., Elbassioni, K., Klau, G.W., Mestre, J.: On tree-constrained matchings and generalizations. In: Proc. of the 38th International Colloquium on Automata, Languages and Programming, pp. 98–109 (2011)

    Google Scholar 

  5. 5.

    Feige, U., Lovász, L.: Two-prover one-round proof systems: their power and their problems. In: Proc. of the 24th Annual ACM Symposium on Theory of Computing, pp. 733–744 (1992)

    Google Scholar 

  6. 6.

    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)

    Book  MATH  Google Scholar 

  7. 7.

    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-dimensional matching. In: Proc. of the 7th International Workshop on Approximation, Randomization, and Combinatorial Optimization, pp. 83–97 (2003)

    Google Scholar 

  8. 8.

    Hoefer, M., Kesselheim, T., Vöcking, B.: Approximation algorithms for secondary spectrum auctions. In: Proc. of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 177–186 (2011)

    Google Scholar 

  9. 9.

    Kortsarz, G.: On the hardness of approximating spanners. Algorithmica 30(3), 432–450 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  11. 11.

    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 Mol. Biol. 4(10) (2009)

  12. 12.

    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    MATH  Google Scholar 

  13. 13.

    Wolsey, L.A., Nemhauser, G.L.: Integer and Combinatorial Optimization. Wiley, New York (1999)

    MATH  Google Scholar 

  14. 14.

    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)

    Article  Google Scholar 

  15. 15.

    Yang, F., Mackey, M.A., Ianzini, F., Gallardo, G., Sonka, M.: Cell segmentation, tracking, and mitosis detection using temporal context. In: Proc. of the 8th International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 302–309 (2005)

    Google Scholar 

  16. 16.

    Ye, Y., Borodin, A.: Elimination graphs. ACM Trans. Algorithms 8(2), 14 (2012)

    Article  MathSciNet  Google Scholar 

  17. 17.

    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 Process. Mag. 23(3), 54–62 (2006)

    Article  Google Scholar 

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Acknowledgements

We thank Axel Mosig for introducing us to the problem and for helpful discussions. We also thank an anonymous reviewer of an earlier version for pointing out the connection between our problem and the work of Bar-Yehuda et al. [2]. Thanks also to Yuk Hei Chan for helpful discussions.

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Correspondence to Julián Mestre.

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An extended abstract of this work appeared in the Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011) [4].

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Canzar, S., Elbassioni, K., Klau, G.W. et al. On Tree-Constrained Matchings and Generalizations. Algorithmica 71, 98–119 (2015). https://doi.org/10.1007/s00453-013-9785-0

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Keywords

  • k-partite matching
  • Rooted trees
  • Approximation algorithms
  • Local ratio technique
  • Inapproximability
  • Computational biology