Abstract
We introduce a strong extended formulation of the convex recoloring problem on a tree, which has an application in analyzing phylogenetic trees. The extended formulation has only a polynomial number of constraints, but dominates the conventional formulation and the exponentially many valid inequalities introduced by Campêlo et al. (Math Progr 156:303–330, 2016). We show that all valid inequalities introduced by Campêlo et al. can be derived from the extended formulation. We also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree. The solution time using the extended formulation is much smaller than that with the conventional formulation. Moreover the extended formulation solves all the problem instances attempted in Campêlo et al. (2016) and larger sized instances at the root node of the branch-and-bound tree without branching.
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References
Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. Theory Comput. Syst. 43, 3–18 (2008)
Campêlo, M., Huiban, C.G., Sampaio, R.M., Wakabayashi, Y.: On the complexity of solving or approximating convex recoloring problems. Lect. Notes Comput. Sci. 7936, 614–625 (2013)
Campêlo, M., Lima, K.R., Moura, P., Wakabayashi, Y.: Polyhedral studies on the CR problem. Electron. Notes Discret. Math. 44, 233–238 (2013)
Campêlo, M., Freire, A.S., Lima, K.R., Moura, P., Wakabayashi, Y.: The convex recoloring problem: polyhedra, facets and computational experiments. Math. Progr. 156, 303–330 (2016)
Chopra, S., Owen, J.L.: Extended formulations for the A-cut problem. Math. Progr. 73, 7–30 (1996)
Chopra, S., Rao, M.R.: The partition problem. Math. Progr. 59, 87–115 (1993)
Chor, B., Fellows, M., Ragan, M., Razgon, I., Rosamond, F., Snir, S.: Connected coloring completion for general graphs: algorithms and complexity. Lect. Comput. Sci. 4598, 75–85 (2007)
Conforti, M., Kaibel, V., Walter, M., Weltge, S.: Subgraph polytopes and independence polytopes of count matroids. Oper. Res. Lett. 43, 457–460 (2015)
Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)
Goemans, M.X.: The Steiner tree polytope and related polyhedra. Math. Progr. 63, 157–182 (1994)
Kaibel, V.: Extended Formulations in Combinatorial Optimization, arXiv:1104.1023v1 [math.CO] 6 Apr (2011), manuscript
Kammer, F., Tholey, T.: The complexity of minimum convex coloring. Discret. Appl. Math. 160, 810–833 (2012)
Kanj, I.A., Kratsch, D.: Convex recoloring revisited: complexity and exact algorithms. Lect. Notes Comput. Sci. 5609, 388–397 (2009)
Korte, B., Lovász, L., Schrader, R.: Greedoids, Algorithms and Combinatorics, vol. 4. Springer, Berlin (1991)
Lima, K.R., Wakabayashi, Y.: Convex recoloring of paths. Discret. Appl. Math. 164, 450–459 (2014)
Moran, S., Snir, S.: Convex recolorings of strings and trees: definitions, hardness results and algorithms, In: Proceedings WADS 2005: 9th International Workshop on Algorithms and Data Structures, pp. 218–232 (2005)
Moran, S., Snir, S.: Efficient approximation of convex recolorings. J. Comput. Syst. Sci. 73, 1078–1089 (2007)
Moran, S., Snir, S.: Convex recolorings of strings and trees: definitions, hardness results and algorithms. J. Comput. Syst. Sci. 74, 850–869 (2008)
Moran, S., Snir, S., Sung, W.K.: Partial convex recolorings of trees and galled networks: tight upper and lower bounds. ACM Trans. Algorithms 7, 42 (2011)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)
Ponta, O., Hüffner, F., Niedermeier, R.: Speeding up dynamic programming for some NP-hard graph recoloring problems, In: Proceedings of the 5th International Conference on Theory and Applications of Models of Computation, TAMC08, pp. 490–501. Springer (2008)
Vanderbeck, F., Wolsey, L.A.: Reformulation and decomposition of integer programs. In: Michael Jünger, T., Liebling, D.N., George, N., William, P., Gerhard, R., Giovanni, R., Laurence, W. (eds.) Fifty Years of Integer Programming 1958–2008, pp. 431–502. Springer, Berlin (2010)
Wang, Y., Buchanan, A., Butenko, S.: On Imposing Connectivity Constraints in Integer Programs. www.optimization-online.org/DB_HTML/2015/02/4768.html
Acknowledgements
We thank Professor Yoshiko Wakabayashi and the other authors of [4] for sharing their data. We are of course grateful for their paper that inspired our work. Mathieu Van Vyve and Bartosz Filipecki were supported by the Interuniversity Attraction Poles Programme P7/36 COMEX of the Belgian Science Policy Office and the Marie Curie ITN “MINO” from the European Commission.
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Chopra, S., Filipecki, B., Lee, K. et al. An extended formulation of the convex recoloring problem on a tree. Math. Program. 165, 529–548 (2017). https://doi.org/10.1007/s10107-016-1094-3
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DOI: https://doi.org/10.1007/s10107-016-1094-3