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.
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Sometimes called primal constraint graph or Gaifman graph.
Probability at least \(1 - 1/|V(G)|^c\) for some constant \(c>1\).
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.
Existence is proven by a counting argument, and it is thus a non-constructive proof. It remains open to find such a family constructively.
Aboulker, P., Fiorini, S., Huynh, T., Macchia, M., Seif, J.: Extension complexity of the correlation polytope. Oper. Res. Lett. 47(1), 47–51 (2019)
Arnborg, S.: Efficient algorithms for combinatorial problems on graphs with bounded decomposability. A survey. BIT Numer. Math. 25, 2–23 (1985)
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)
Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebr. Discret. Methods 8(2), 277–284 (1987)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)
Arora, S., Barak, B.: Computational Complexity—A Modern Approach. Cambridge University Press, Cambridge (2009)
Avis, D., Tiwary, H.R.: A generalization of extension complexity that captures p. Inf. Process. Lett. 115(6–8), 588–593 (2015a)
Avis, D., Tiwary, H.R.: On the extension complexity of combinatorial polytopes. Math. Program. 153(1), 95–115 (2015b)
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)
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)
Bern, M., Lawler, E., Wong, A.: Linear-time computation of optimal subgraphs of decomposable graphs. J. Algorithms 8(2), 216–235 (1987)
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)
Bienstock, D., Muñoz, G.: LP formulations for polynomial optimization problems. SIAM J. Optim. 28(2), 1121–1150 (2018)
Bienstock, D., Özbay, N.: Tree-width and the Sherali–Adams operator. Discret. Optim. 1(1), 13–21 (2004)
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)
Braun, G., Fiorini, S., Pokutta, S., Steurer, D.: Approximation limits of linear programs (beyond hierarchies). Math. Oper. Res. 40(3), 756–772 (2015)
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
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)
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)
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)
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)
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)
Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. J. ACM (JACM) 63(5), 40 (2016)
Chuzhoy, J.: Improved bounds for the excluded grid theorem (2016). ArXiv preprint arXiv:1602.02629
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
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)
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)
Dechter, R., Pearl, J.: Tree clustering for constraint networks (research note). Artif. Intell. 38, 353–366 (1989)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, New York (2012)
Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program. 153(1), 133–177 (2015)
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
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)
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)
Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985). https://doi.org/10.1145/4221.4225
Fulkerson, D.R., Gross, O.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)
Gajarský, J., Hliněný, P., Tiwary, H.R.: Parameterized extension complexity of independent set and related problems. Discret. Appl. Math. 248, 56–67 (2017)
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)
Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38(2), 248–264 (2013)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM (JACM) 54(1), 1 (2007)
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)
Halin, R.: S-functions for graphs. J. Geom. 8(1–2), 171–186 (1976)
Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001). https://doi.org/10.1145/502090.502098
Kolman, P., Koutecký, M.: Extended formulation for csp that is compact for instances of bounded treewidth. Electron. J. Combin. 22(4), P4–30 (2015)
Kolman, P., Koutecký, M., Tiwary, H.R.: Extension complexity, mso logic, and treewidth (2015). ArXiv preprint arXiv:1507.04907
Lasserre, J.: Convergent SDP relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)
Laurent, M.: Sum of squares, moment matrices and optimization over polynomials. IMA, pp. 1–147 (2010)
Lauritzen, S.L.: Graphical Models. Oxford University Press, Oxford (1996)
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)
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)
Marx, D.: Can you beat treewidth? Theory Comput. 6, 85–112 (2010)
Muñoz, G.: Integer programming techniques for polynomial optimization. PhD thesis, Columbia University (2017). https://doi.org/10.7916/D82F812G
Pearl, J.: Reverend bayes on inference engines: a distributed hierarchical approach. In: Proceedings of the National Conference on Artificial Intelligence, pp. 133–136 (1982)
Robertson, N., Seymour, P.: Graph minors III. Planar tree-width. J. Combin. Theory Ser. B 36(1), 49–64 (1984)
Robertson, N., Seymour, P.: Graph minors II: algorithmic aspects of tree-width. J. Algorithm 7, 309–322 (1986)
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
Rothvoß, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Program. 142(1–2), 255–268 (2013)
Rothvoß, T.: The matching polytope has exponential extension complexity. J. ACM (JACM) 64(6), 41 (2017)
Tamassia, R., Tollis, I.G.: Planar grid embedding in linear time. IEEE Trans. Circuits Syst. 36(9), 1230–1234 (1989)
Tiwary, H.R., Weltge, S., Zenklusen, R.: Extension complexities of Cartesian products involving a pyramid (2017). ArXiv preprint arXiv:1702.01959
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)
Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1, 1–305 (2008)
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)
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
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)
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
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).
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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
- Structured sparsity
- Semidefinite extension complexity
- Approximations of QCQPs
Mathematics Subject Classification