, Volume 71, Issue 1, pp 98–119 | Cite as

On Tree-Constrained Matchings and Generalizations

  • Stefan Canzar
  • Khaled Elbassioni
  • Gunnar W. Klau
  • Julián MestreEmail author


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.


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



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.


  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) CrossRefzbMATHMathSciNetGoogle 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) CrossRefzbMATHMathSciNetGoogle 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) CrossRefGoogle 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) CrossRefGoogle 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) CrossRefzbMATHGoogle 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) CrossRefGoogle 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) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge University Press, Cambridge (2011) CrossRefzbMATHGoogle 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) Google Scholar
  12. 12.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986) zbMATHGoogle Scholar
  13. 13.
    Wolsey, L.A., Nemhauser, G.L.: Integer and Combinatorial Optimization. Wiley, New York (1999) zbMATHGoogle 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) CrossRefGoogle 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) CrossRefMathSciNetGoogle 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) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Stefan Canzar
    • 1
  • Khaled Elbassioni
    • 2
  • Gunnar W. Klau
    • 1
  • Julián Mestre
    • 3
    Email author
  1. 1.Centrum Wiskunde & InformaticaLife Sciences GroupAmsterdamThe Netherlands
  2. 2.Masdar Institute of Science and TechnologyAbu DhabiUnited Arab Emirates
  3. 3.School of Information TechnologiesThe University of SydneySydneyAustralia

Personalised recommendations