A Benders Approach to the Minimum Chordal Completion Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9075)


This paper introduces an integer programming approach to the minimum chordal completion problem. This combinatorial optimization problem, although simple to pose, presents considerable computational difficulties and has been tackled mostly by heuristics. In this paper, an integer programming approach based on Benders decomposition is presented. Computational results show that the improvement in solution times over a simple branch-and-bound algorithm is substantial. The results also indicate that the value of the solutions obtained by a state-of-the-art heuristic can be in some cases significantly far away from the previously unknown optimal solutions obtained via the Benders approach.


Master Problem Chordal Graph Bender Decomposition Search Node Warm Start 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. J. ACM 30(3), 479–513 (1983). http://doi.acm.org/10.1145/2402.322389 CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Benders, J.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4(1), 238–252 (1962). http://dx.doi.org/10.1007/BF01386316 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berry, A., Bordat, J.P., Heggernes, P., Simonet, G., Villanger, Y.: A wide-range algorithm for minimal triangulation from an arbitrary ordering. Journal of Algorithms 58(1), 33–66 (2006). http://www.sciencedirect.com/science/article/pii/S0196677404001142 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Berry, A., Heggernes, P., Simonet, G.: The minimum degree heuristic and the minimal triangulation process. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 58–70. Springer, Heidelberg (2003). http://www.dx.doi.org/10.1007/978-3-540-39890-5_6 CrossRefGoogle Scholar
  5. 5.
    Blair, J.R., Heggernes, P., Telle, J.A.: A practical algorithm for making filled graphs minimal. Theoretical Computer Science 250(12), 125–141 (2001). http://www.sciencedirect.com/science/article/pii/S0304397599001267 CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–23 (1993)MATHMathSciNetGoogle Scholar
  7. 7.
    Bodlaender, H.L., Kloks, T., Kratsch, D., Mueller, H.: Treewidth and minimum fill-in on d-trapezoid graphs (1998)Google Scholar
  8. 8.
    Bodlaender, H.L., Heggernes, P., Villanger, Y.: Faster parameterized algorithms for Minimum Fill-In. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 282–293. Springer, Heidelberg (2008). http://dx.doi.org/10.1007/978-3-540-92182-0_27 CrossRefGoogle Scholar
  9. 9.
    Broersma, H., Dahlhaus, E., Kloks, T.: Algorithms for the treewidth and minimum fill-in of hhd-free graphs. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 109–117. Springer, Heidelberg (1997). http://dx.doi.org/10.1007/BFb0024492 CrossRefGoogle Scholar
  10. 10.
    Chandran, L.S., Ibarra, L., Ruskey, F., Sawada, J.: Generating and characterizing the perfect elimination orderings of a chordal graph. Theor. Comput. Sci. 307(2), 303–317 (2003). http://dx.doi.org/10.1016/S0304-3975(03)00221-4 CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Chang, M.S.: Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. In: Nagamochi, H., Suri, S., Igarashi, Y., Miyano, S., Asano, T. (eds.) ISAAC 1996. LNCS, vol. 1178, pp. 146–155. Springer, Heidelberg (1996). http://dx.doi.org/10.1007/BFb0009490 CrossRefGoogle Scholar
  12. 12.
    Chung, F., Mumford, D.: Chordal completions of planar graphs. Journal of Combinatorial Theory, Series B 62(1), 96–106 (1994). http://www.sciencedirect.com/science/article/pii/S0095895684710562 CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. In: Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1737–1746. SIAM (2012), http://dl.acm.org/citation.cfm?id=2095116.2095254
  14. 14.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific Journal of Mathematics 15(3), 835–855 (1965). http://projecteuclid.org/euclid.pjm/1102995572 CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979) MATHGoogle Scholar
  16. 16.
    George, A., Liu, W.H.: The evolution of the minimum degree ordering algorithm. SIAM Rev. 31(1), 1–19 (1989). http://dx.doi.org/10.1137/1031001 CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Golumbic, M., Kaplan, H., Shamir, R.: Graph sandwich problems. Journal of Algorithms 19(3), 449–473 (1995). http://www.sciencedirect.com/science/article/pii/S0196677485710474 CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra and its Applications 58, 109–124 (1984). http://www.sciencedirect.com/science/article/pii/0024379584902076 CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Heggernes, P.: Minimal triangulations of graphs: A survey. Discrete Mathematics 306(3), 297–317 (2006). http://www.sciencedirect.com/science/article/pii/S0012365X05006060 CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Heggernes, P., Eisenstat, S.C., Kumfert, G., Pothen, A.: The computational complexity of the minimum degree algorithm (2001)Google Scholar
  21. 21.
    Hooker, J., Ottosson, G.: Logic-based benders decomposition. Mathematical Programming 96(1), 33–60 (2003). http://dx.doi.org/10.1007/s10107-003-0375-9 MATHMathSciNetGoogle Scholar
  22. 22.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping (extended abstract). SIAM J. Comput 28, 780–791 (1994)MathSciNetGoogle Scholar
  23. 23.
    Kim, S., Kojima, M., Mevissen, M., Yamashita, M.: Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion. Mathematical Programming 129(1), 33–68 (2011). http://dx.doi.org/10.1007/s10107-010-0402-6 CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kloks, T., Kratsch, D., Wong, C.: Minimum fill-in on circle and circular-arc graphs. Journal of Algorithms 28(2), 272–289 (1998). http://www.sciencedirect.com/science/article/pii/S0196677498909361 CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lekkeikerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51, 45–64 (1962)MathSciNetGoogle Scholar
  26. 26.
    Mezzini, M., Moscarini, M.: Simple algorithms for minimal triangulation of a graph and backward selection of a decomposable markov network. Theoretical Computer Science 411(79), 958–966 (2010). http://www.sciencedirect.com/science/article/pii/S030439750900735X CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion ii: implementation and numerical results. Mathematical Programming 95(2), 303–327 (2003). http://dx.doi.org/10.1007/s10107-002-0351-9 CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nguyen, T.H., Bui, T.: Graph coloring benchmark instances. http://www.cs.hbg.psu.edu/txn131/graphcoloring.html (accessed: 14 July 2014)
  29. 29.
    Peyton, B.W.: Minimal orderings revisited. SIAM J. Matrix Anal. Appl. 23(1), 271–294 (2001). http://dx.doi.org/10.1137/S089547989936443X CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Rose, D., Tarjan, R., Lueker, G.: Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing 5(2), 266–283 (1976). http://dx.doi.org/10.1137/0205021 CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Rose, D.J.: A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Graph Theory and Computing, pp. 183–217. Academic Press (1972), http://www.sciencedirect.com/science/article/pii/B9781483231877500180
  32. 32.
    Sokhn, N., Baltensperger, R., Bersier, L.-F., Hennebert, J., Ultes-Nitsche, U.: Identification of chordless cycles in ecological networks. In: Glass, K., Colbaugh, R., Ormerod, P., Tsao, J. (eds.) Complex 2012. LNICST, vol. 126, pp. 316–324. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  33. 33.
    Spinrad, J., Brandstdt, A., Stewart, L.: Bipartite permutation graphs. Discrete Applied Mathematics 18(3), 279–292 (1987). http://www.sciencedirect.com/science/article/pii/S0166218X87800033 CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984). http://dx.doi.org/10.1137/0213035 CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Uno, T., Satoh, H.: An efficient algorithm for enumerating chordless cycles and chordless paths. CoRR abs/1404.7610 (2014), http://arxiv.org/abs/1404.7610
  36. 36.
    Yannakakis, M.: Computing the minimum fill-in is np-complete. SIAM Journal on Algebraic Discrete Methods 2(1), 77–79 (1981)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of ConnecticutStamfordUSA
  2. 2.Mitsubishi Electric Research LabsCambridgeUSA

Personalised recommendations