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# Extension Complexity, MSO Logic, and Treewidth

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LIPIcs.SWAT.2016.18.pdf
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## Cite As

Petr Kolman, Martin Koutecký, and Hans Raj Tiwary. Extension Complexity, MSO Logic, and Treewidth. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SWAT.2016.18

## Abstract

We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau. In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs.
##### Keywords
• Extension Complexity
• FPT
• Courcelle's Theorem
• MSO Logic

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## References

1. S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, June 1991.
2. D. Avis and H. R. Tiwary. On the extension complexity of combinatorial polytopes. In Proc. ICALP(1), pages 57-68, 2013.
3. D. Bienstock and G. Munoz. LP approximations to mixed-integer polynomial optimization problems. ArXiv e-prints, January 2015. URL: http://arxiv.org/abs/1501.00288.
4. H. L. Bodlaender. A linear time algorithm for finding tree-decompositions of small treewidth. In Proc. STOC, pages 226-234, 1993.
5. H. L. Bodlaender. Treewidth: characterizations, applications, and computations. In Proc. of WG, volume 4271 of LNCS, pages 1-14. Springer, 2006.
6. G. Braun, S. Fiorini, S. Pokutta, and D. Steurer. Approximation limits of linear programs (beyond hierarchies). Math. Oper. Res., 40(3):756-772, 2015.
7. G. Braun, R. Jain, T. Lee, and S. Pokutta. Information-theoretic approximations of the nonnegative rank. Electronic Colloquium on Computational Complexity, 20:158, 2013.
8. A. Buchanan and S. Butenko. Tight extended formulations for independent set, 2014. Available on Optimization Online. URL: http://www.optimization-online.org/DB_HTML/2014/09/4540.html.
9. M. Conforti, G. Cornuéjols, and G. Zambelli. Extended formulations in combinatorial optimization. Annals of Operations Research, 204(1):97-143, 2013.
10. M. Conforti and K. Pashkovich. The projected faces property and polyhedral relations. Mathematical Programming, pages 1-12, 2015.
11. B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12-75, 1990.
12. B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In Proc. of WG, volume 1517 of LNCS, pages 125-150. Springer, 1998.
13. B. Courcelle and M. Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science, 109(1-2):49-82, 1 March 1993.
14. Y. Faenza, S. Fiorini, R. Grappe, and H. R. Tiwary. Extended formulations, nonnegative factorizations, and randomized com. protocols. Math. Program., 153(1):75-94, 2015.
15. S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, and Ronald de Wolf. Exponential lower bounds for polytopes in combinatorial optimization. J. ACM, 62(2):17, 2015.
16. G. Gottlob, R. Pichler, and F. Wei. Monadic datalog over finite structures with bounded treewidth. In Proc. PODS, pages 165-174, 2007.
17. B. Grünbaum. Convex Polytopes. Wiley Interscience Publ., London, 1967.
18. V. Kaibel. Extended formulations in combinatorial optimization. Optima, 85:2-7, 2011.
19. V. Kaibel and A. Loos. Branched polyhedral systems. In Proc. IPCO, volume 6080 of LNCS, pages 177-190. Springer, 2010.
20. V. Kaibel and K. Pashkovich. Constructing extended formulations from reflection relations. In Proc. IPCO, volume 6655 of LNCS, pages 287-300. Springer, 2011.
21. L. Kaiser, M. Lang, S. Leßenich, and Ch. Löding. A Unified Approach to Boundedness Properties in MSO. In Proc. of CSL, volume 41 of LIPIcs, pages 441-456, 2015.
22. T. Kloks. Treewidth: Computations and Approximations, volume 842 of LNCS. Springer, 1994.
23. J. Kneis, A. Langer, and P. Rossmanith. Courcelle’s theorem - A game-theoretic approach. Discrete Optimization, 8(4):568-594, 2011.
24. P. G. Kolaitis and M. Y. Vardi. Conjunctive-query containment and constraint satisfaction. In Proc. PODS, 1998.
25. P. Kolman and M. Koutecký. Extended formulation for CSP that is compact for instances of bounded treewidth. Electr. J. Comb., 22(4):P4.30, 2015.
26. S. Kreutzer. Algorithmic meta-theorems. In Proc. of IWPEC, volume 5018 of LNCS, pages 10-12. Springer, 2008.
27. A. Langer, F. Reidl, P. Rossmanith, and S. Sikdar. Practical algorithms for MSO model-checking on tree-decomposable graphs. Computer Science Review, 13-14:39-74, 2014.
28. M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, pages 157-270. Springer, 2009.
29. J. R. Lee, P. Raghavendra, and D. Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proc. STOC, pages 567-576, 2015.
30. L. Libkin. Elements of Finite Model Theory. Springer, Berlin, 2004.
31. F. Margot. Composition de polytopes combinatoires: une approche par projection. PhD thesis, École polytechnique fédérale de Lausanne, 1994.
32. R. K. Martin, R. L. Rardin, and B. A. Campbell. Polyhedral characterization of discrete dynamic programming. Oper. Res., 38(1):127-138, February 1990.
33. M. Sellmann. The polytope of tree-structured binary constraint satisfaction problems. In Proc. CPAIOR, volume 5015 of LNCS, pages 367-371. Springer, 2008.
34. M. Sellmann, L. Mercier, and D. H. Leventhal. The linear programming polytope of binary constraint problems with bounded tree-width. In Proc. CPAIOR, volume 4510 of LNCS, pages 275-287. Springer, 2007.
35. F. Vanderbeck and L. A. Wolsey. Reformulation and decomposition of integer programs. In 50 Years of Integer Programming 1958-2008, pages 431-502. Springer, 2010.
36. L. A. Wolsey. Using extended formulations in practice. Optima, 85:7-9, 2011.
37. M. Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441-466, 1991.
38. G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer, 1995.
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