Skip to main content

New limits of treewidth-based tractability in optimization

Abstract

Sparse structures are frequently sought when pursuing tractability in optimization problems. They are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present. An example of this type of structure is given by treewidth: a graph theoretical parameter that measures how “tree-like” a graph is. This parameter has been used for decades for analyzing the complexity of various optimization problems and for obtaining tractable algorithms for problems where this parameter is bounded. The goal of this work is to contribute to the understanding of the limits of the treewidth-based tractability in optimization. Our results are as follows. First, we prove that, in a certain sense, the already known positive results on extension complexity based on low treewidth are the best possible. Secondly, under mild assumptions, we prove that treewidth is the only graph-theoretical parameter that yields tractability for a wide class of optimization problems, a fact well known in Graphical Models in Machine Learning and in Constraint Satisfaction Problems, which here we extend to an approximation setting in Optimization.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Sometimes called primal constraint graph or Gaifman graph.

  2. 2.

    Probability at least \(1 - 1/|V(G)|^c\) for some constant \(c>1\).

  3. 3.

    By equivalent, we mean that feasible solutions of one optimization problem can be mapped to feasible solutions of the other with the same objective value, and vice-versa.

  4. 4.

    Existence is proven by a counting argument, and it is thus a non-constructive proof. It remains open to find such a family constructively.

References

  1. 1.

    Aboulker, P., Fiorini, S., Huynh, T., Macchia, M., Seif, J.: Extension complexity of the correlation polytope. Oper. Res. Lett. 47(1), 47–51 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Arnborg, S.: Efficient algorithms for combinatorial problems on graphs with bounded decomposability. A survey. BIT Numer. Math. 25, 2–23 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems on graphs embedded in k-trees. Tech. Rep. TRITA-NA-8404, The Royal Institute of Technology, Stockholm (1984)

  4. 4.

    Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebr. Discret. Methods 8(2), 277–284 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Arora, S., Barak, B.: Computational Complexity—A Modern Approach. Cambridge University Press, Cambridge (2009)

    MATH  Book  Google Scholar 

  7. 7.

    Avis, D., Tiwary, H.R.: A generalization of extension complexity that captures p. Inf. Process. Lett. 115(6–8), 588–593 (2015a)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Avis, D., Tiwary, H.R.: On the extension complexity of combinatorial polytopes. Math. Program. 153(1), 95–115 (2015b)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bazzi, A., Fiorini, S., Pokutta, S., Svensson, O.: No small linear program approximates vertex cover within a factor 2–e. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, pp. 1123–1142 (2015)

  10. 10.

    Bazzi, A., Fiorini, S., Pokutta, S., Svensson, O.: No small linear program approximates vertex cover within a factor 2–e. In: Mathematics of Operations Research (2018) (to appear)

  11. 11.

    Bern, M., Lawler, E., Wong, A.: Linear-time computation of optimal subgraphs of decomposable graphs. J. Algorithms 8(2), 216–235 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Bienstock, D., Langston, M.A.: Chapter 8 algorithmic implications of the graph minor theorem. In: Ball, M.O., Magnant, C.M., Nemhauser, G. (eds.) Network Models. Handbooks in Operations Research and Management Science, pp. 481–502. Elsevier, New York (1995)

    Chapter  Google Scholar 

  13. 13.

    Bienstock, D., Muñoz, G.: LP formulations for polynomial optimization problems. SIAM J. Optim. 28(2), 1121–1150 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Bienstock, D., Özbay, N.: Tree-width and the Sherali–Adams operator. Discret. Optim. 1(1), 13–21 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bodlaender, H.: Dynamic programming on graphs with bounded treewidth. In: Lepistö, T., Salomaa, A. (eds.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 317, pp. 105–118. Springer, Berlin (1988)

    Chapter  Google Scholar 

  16. 16.

    Braun, G., Fiorini, S., Pokutta, S., Steurer, D.: Approximation limits of linear programs (beyond hierarchies). Math. Oper. Res. 40(3), 756–772 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Braun, G., Fiorini, S., Pokutta, S.: Average case polyhedral complexity of the maximum stable set problem. Math. Program. 160(1–2), 407–431 (2016a). https://doi.org/10.1007/s10107-016-0989-3

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Braun, G., Pokutta, S., Roy, A.: Strong reductions for extended formulations. In: International Conference on Integer Programming and Combinatorial Optimization, Springer, New York, pp. 350–361 (2016b)

  19. 19.

    Briët, J., Dadush, D., Pokutta, S.: On the existence of 0/1 polytopes with high semidefinite extension complexity. In: European Symposium on Algorithms, Springer, New York, pp. 217–228 (2013)

  20. 20.

    Briët, J., Dadush, D., Pokutta, S.: On the existence of 0/1 polytopes with high semidefinite extension complexity. Math. Program. 153(1), 179–199 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Brown, D.J., Fellows, M.R., Langston, M.A.: Polynomial-time self-reducibility: theoretical motivations and practical results. Int. J. Comput. Math. 31(1–2), 1–9 (1989)

    MATH  Article  Google Scholar 

  22. 22.

    Chandrasekaran, V., Srebro, N., Harsha, P.: Complexity of inference in graphical models. In: Proceedings of the Twenty-Fourth Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-08), AUAI Press, Corvallis, Oregon, pp. 70–78 (2008)

  23. 23.

    Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. J. ACM (JACM) 63(5), 40 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Chuzhoy, J.: Improved bounds for the excluded grid theorem (2016). ArXiv preprint arXiv:1602.02629

  25. 25.

    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. In: International Workshop on Graph-Theoretic Concepts in Computer Science, Springer, New York, pp. 13–24 (1991)

  27. 27.

    Cunningham, W.H,. Geelen, J.: On integer programming and the branch-width of the constraint matrix. In: International Conference on Integer Programming and Combinatorial Optimization, Springer, New York, pp. 158–166 (2007)

  28. 28.

    Dechter, R., Pearl, J.: Tree clustering for constraint networks (research note). Artif. Intell. 38, 353–366 (1989)

    MATH  Article  Google Scholar 

  29. 29.

    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, New York (2012)

    MATH  Google Scholar 

  30. 30.

    Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program. 153(1), 133–177 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Feige, U., Jozeph, S.: Demand queries with preprocessing. In: Automata, Languages, and Programming—41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8–11, 2014, Proceedings, Part I, pp. 477–488, (2014) https://doi.org/10.1007/978-3-662-43948-7_40

  32. 32.

    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., De Wolf, R.: Linear versus semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, ACM, pp. 95–106 (2012)

  33. 33.

    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., Wolf, R.D.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM (JACM) 62(2), 17 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985). https://doi.org/10.1145/4221.4225

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Fulkerson, D.R., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Gajarský, J., Hliněný, P., Tiwary, H.R.: Parameterized extension complexity of independent set and related problems. Discret. Appl. Math. 248, 56–67 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Göös, M., Jain, R., Watson, T.: Extension complexity of independent set polytopes. In: 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, pp. 565–572 (2016)

  38. 38.

    Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38(2), 248–264 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM (JACM) 54(1), 1 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Guibas, L.J., Hershberger, J.E., Mitchell, J.S., Snoeyink, J.S.: Approximating polygons and subdivisions with minimum link paths. In: International Symposium on Algorithms, Springer, pp. 151–162 (1991)

  41. 41.

    Halin, R.: S-functions for graphs. J. Geom. 8(1–2), 171–186 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001). https://doi.org/10.1145/502090.502098

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Kolman, P., Koutecký, M.: Extended formulation for csp that is compact for instances of bounded treewidth. Electron. J. Combin. 22(4), P4–30 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Kolman, P., Koutecký, M., Tiwary, H.R.: Extension complexity, mso logic, and treewidth (2015). ArXiv preprint arXiv:1507.04907

  45. 45.

    Lasserre, J.: Convergent SDP relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Laurent, M.: Sum of squares, moment matrices and optimization over polynomials. IMA, pp. 1–147 (2010)

  47. 47.

    Lauritzen, S.L.: Graphical Models. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  48. 48.

    Lee, J.R., Raghavendra, P., Steurer, D.: Lower bounds on the size of semidefinite programming relaxations. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, ACM, pp. 567–576 (2015)

  49. 49.

    Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp 777–789 (2011)

  50. 50.

    Marx, D.: Can you beat treewidth? Theory Comput. 6, 85–112 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Muñoz, G.: Integer programming techniques for polynomial optimization. PhD thesis, Columbia University (2017). https://doi.org/10.7916/D82F812G

  52. 52.

    Pearl, J.: Reverend bayes on inference engines: a distributed hierarchical approach. In: Proceedings of the National Conference on Artificial Intelligence, pp. 133–136 (1982)

  53. 53.

    Robertson, N., Seymour, P.: Graph minors III. Planar tree-width. J. Combin. Theory Ser. B 36(1), 49–64 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Robertson, N., Seymour, P.: Graph minors II: algorithmic aspects of tree-width. J. Algorithm 7, 309–322 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62(2), 323–348 (1994). https://doi.org/10.1006/jctb.1994.1073

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Rothvoß, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Program. 142(1–2), 255–268 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Rothvoß, T.: The matching polytope has exponential extension complexity. J. ACM (JACM) 64(6), 41 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Tamassia, R., Tollis, I.G.: Planar grid embedding in linear time. IEEE Trans. Circuits Syst. 36(9), 1230–1234 (1989)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Tiwary, H.R., Weltge, S., Zenklusen, R.: Extension complexities of Cartesian products involving a pyramid (2017). ArXiv preprint arXiv:1702.01959

  60. 60.

    Wainwright, M.J., Jordan, M.I.: Treewidth-based conditions for exactness of the Sherali–Adams and Lasserre relaxations. Tech. Rep. 671, University of California (2004)

  61. 61.

    Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1, 1–305 (2008)

    MATH  Article  Google Scholar 

  62. 62.

    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Wang, C., Liu, T., Cui, P., Xu, K.: A note on treewidth in random graphs. In: Combinatorial Optimization and Applications—5th International Conference, COCOA 2011, Zhangjiajie, China, August 4–6, 2011. Proceedings, pp. 491–499 (2011). https://doi.org/10.1007/978-3-642-22616-8_38

  64. 64.

    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  65. 65.

    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007). https://doi.org/10.4086/toc.2007.v003a006

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous reviewers whose suggestions greatly helped improving this article. Research reported in this paper was partially supported by NSF CAREER award CMMI-1452463 and by the Institute for Data Valorisation (IVADO).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gonzalo Muñoz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Faenza, Y., Muñoz, G. & Pokutta, S. New limits of treewidth-based tractability in optimization. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01563-5

Download citation

Keywords

  • Treewidth
  • Structured sparsity
  • Semidefinite extension complexity
  • Approximations of QCQPs

Mathematics Subject Classification

  • 90C05
  • 90C22
  • 90C35