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Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach

Authors Lars Jaffke , Mateus de Oliveira Oliveira, Hans Raj Tiwary

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Lars Jaffke
  • University of Bergen, Norway
Mateus de Oliveira Oliveira
  • University of Bergen, Norway
Hans Raj Tiwary
  • Charles University, Prague, Czech Republic


We thank Manuel Aprile, Laszlo Babai, Peter Cameron, Michael Fellows and Samuel Fiorini for valuable comments and suggestions. We thank Michel Goemans, Kanstantsin Pashkovich and Stefan Weltge for answering some of our questions by email.

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Lars Jaffke, Mateus de Oliveira Oliveira, and Hans Raj Tiwary. Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 50:1-50:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


It can be shown that each permutation group G ⊑ 𝕊_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, 𝕊_n itself can be embedded in the n-clique K_n, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group G⊑ 𝕊_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G ⊑ 𝕊_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΔlogΔ)}⋅ m^{O(k)}. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ω(n)} lower bound on the grammar complexity of the symmetric group 𝕊_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of 𝕊_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 1 and maximum degree o(n/log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic language theory
  • Permutation Groups
  • Context Free Grammars
  • Extension Complexity
  • Graph Embedding Complexity


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  1. Miklós Ajtai, János Komlós, and Endre Szemerédi. Sorting in logn parallel steps. Combinatorica, 3(1):1-19, 1983. Google Scholar
  2. Peter RJ Asveld. Generating all permutations by context-free grammars in Chomsky normal form. Theoretical Computer Science, 354(1):118-130, 2006. Google Scholar
  3. Peter RJ Asveld. Generating all permutations by context-free grammars in Greibach normal form. Theoretical Computer Science, 409(3):565-577, 2008. Google Scholar
  4. David Avis and Hans Raj Tiwary. On the extension complexity of combinatorial polytopes. Mathematical Programming, 153(1):95-115, 2015. Google Scholar
  5. László Babai. Representation of permutation groups by graphs. Colloquia Mathematica Societatis Janos Bolyai, 4:55-80, 1969. Google Scholar
  6. László Babai. Automorphism groups of graphs and edge-contraction. Discrete Mathematics, 8(1):13-20, 1974. Google Scholar
  7. László Babai. On the abstract group of automorphisms. In HNV Temperley, editor, Combinatorics, volume 52 of London Mathematical Society Lecture Note Series, pages 1-40. Cambridge University Press, 1981. Google Scholar
  8. László Babai. Automorphism groups, isomorphism, and reconstruction. In Handbook of Combinatorics, pages 1447-1540. North-Holland-Elsevier, 1995. Google Scholar
  9. Egon Balas and William Pulleyblank. The perfectly matchable subgraph polytope of a bipartite graph. Networks, 13(4):495-516, 1983. Google Scholar
  10. Francisco Barahona. On cuts and matchings in planar graphs. Mathematical Programming, 60(1-3):53-68, 1993. Google Scholar
  11. Aharon Ben-Tal and Arkadi Nemirovski. On polyhedral approximations of the second-order cone. Mathematics of Operations Research, 26(2):193-205, 2001. Google Scholar
  12. IZ Bouwer. Section graphs for finite permutation groups. Journal of Combinatorial Theory, 6(4):378-386, 1969. Google Scholar
  13. Gábor Braun, Samuel Fiorini, Sebastian Pokutta, and David Steurer. Approximation limits of linear programs (beyond hierarchies). Mathematics of Operations Research, 40(3):756-772, 2015. Google Scholar
  14. Kevin King Hin Cheung. Subtour Elimination Polytopes and Graphs of Inscribable Type. PhD thesis, University of Waterloo, 2003. Google Scholar
  15. Michele Conforti, Marco Di Summa, Friedrich Eisenbrand, and Laurence A Wolsey. Network formulations of mixed-integer programs. Mathematics of Operations Research, 34(1):194-209, 2009. Google Scholar
  16. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  17. Michel M Deza and Monique Laurent. Geometry of Cuts and Metrics. Springer, 1997. Google Scholar
  18. Keith Ellul, Bryan Krawetz, Jeffrey Shallit, and Ming-wei Wang. Regular expressions: New results and open problems. Journal of Automata, Languages and Combinatorics, 9(2/3):233-256, 2004. Google Scholar
  19. Yuri Faenza and Volker Kaibel. Extended formulations for packing and partitioning orbitopes. Mathematics of Operations Research, 34(3):686-697, 2009. Google Scholar
  20. Yuval Filmus. Lower bounds for context-free grammars. Information Processing Letters, 111(18):895-898, 2011. Google Scholar
  21. Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary, and Ronald De Wolf. Exponential lower bounds for polytopes in combinatorial optimization. Journal of the ACM, 62(2):17, 2015. Google Scholar
  22. Ian Glaister and Jeffrey Shallit. A lower bound technique for the size of nondeterministic finite automata. Information Processing Letters, 59(2):75-77, 1996. Google Scholar
  23. Michel X Goemans. Smallest compact formulation for the permutahedron. Mathematical Programming, 153(1):5-11, 2015. Google Scholar
  24. Massimiliano Goldwurm, Beatrice Palano, and Massimo Santini. On the circuit complexity of random generation problems for regular and context-free languages. In Afonso Ferreira and Horst Reichel, editors, Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2001, volume 2010 of LNCS, pages 305-316. Springer, 2001. Google Scholar
  25. Hermann Gruber, Markus Holzer, and Simon Wolfsteiner. On minimal grammar problems for finite languages. In Mizuho Hoshi and Shinnosuke Seki, editors, Proceedings of the 22nd International Conference on Developments in Language Theory, DLT 2018, volume 11088 of LNCS, pages 342-353. Springer, 2018. Google Scholar
  26. James E Humphreys. Reflection Groups and Coxeter Groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1992. Google Scholar
  27. Volker Kaibel and Andreas Loos. Branched polyhedral systems. In Friedrich Eisenbrand and F. Bruce Shepherd, editors, Proceedings of the 14th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2010, volume 6080 of LNCS, pages 177-190. Springer, 2010. Google Scholar
  28. Volker Kaibel and Kanstantsin Pashkovich. Constructing extended formulations from reflection relations. In Michael Jünger and Gerhard Reinelt, editors, Facets of Combinatorial Optimization, pages 77-100. Springer, 2013. Google Scholar
  29. Volker Kaibel and Stefan Weltge. A short proof that the extension complexity of the correlation polytope grows exponentially. Discrete & Computational Geometry, 53(2):397-401, 2015. Google Scholar
  30. Martin W Liebeck. On graphs whose full automorphism group is an alternative group or a finite classical group. Proceedings of the London Mathematical Society, 3(2):337-362, 1983. Google Scholar
  31. Antonio Molina Lovett and Jeffrey O. Shallit. Optimal regular expressions for permutations. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, Proceedings of the 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, volume 132 of LIPIcs, pages 121:1-121:12. Schloss Dagstuhl, 2019. Google Scholar
  32. Stefan Mengel. Arithmetic branching programs with memory. In Krishnendu Chatterjee and Jirí Sgall, editors, Proceedings of the 38th International Symposium Mathematical Foundations of Computer Science, MFCS 2013, volume 8087 of LNCS, pages 667-678. Springer, 2013. Google Scholar
  33. Gilles Pesant, Claude-Guy Quimper, Louis-Martin Rousseau, and Meinolf Sellmann. The polytope of context-free grammar constraints. In Willem Jan van Hoeve and John N. Hooker, editors, Proceedings of the 6th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR 2009, volume 5547 of LNCS, pages 223-232, 2009. Google Scholar
  34. Sebastian Pokutta and Mathieu Van Vyve. A note on the extension complexity of the knapsack polytope. Operations Research Letters, 41(4):347-350, 2013. Google Scholar
  35. William R Pulleyblank and Bruce Shepherd. Formulations for the stable set polytope. In Giovanni Rinaldi and Laurence A. Wolsey, editors, Proceedings of the 3rd Conference on Integer Programming and Combinatorial Optimization, IPCO 1993, pages 267-279, 1993. Google Scholar
  36. Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B, 52(2):153-190, 1991. Google Scholar
  37. Thomas Rothvoß. Some 0/1 polytopes need exponential size extended formulations. Mathematical Programming, 142(1-2):255-268, 2013. Google Scholar
  38. Stefan Weltge. Erweiterte Formulierungen für das Alternaeder. Diplomarbeit, University of Magdeburg, Germany, 2012. Google Scholar
  39. Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441-466, 1991. Google Scholar
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