Size-Treewidth Tradeoffs for Circuits Computing the Element Distinctness Function

Author Mateus de Oliveira Oliveira

Thumbnail PDF


  • Filesize: 0.65 MB
  • 14 pages

Document Identifiers

Author Details

Mateus de Oliveira Oliveira

Cite AsGet BibTex

Mateus de Oliveira Oliveira. Size-Treewidth Tradeoffs for Circuits Computing the Element Distinctness Function. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


In this work we study the relationship between size and treewidth of circuits computing variants of the element distinctness function. First, we show that for each n, any circuit of treewidth t computing the element distinctness function delta_n:{0,1}^n -> {0,1} must have size at least Omega((n^2)/(2^{O(t)}*log(n))). This result provides a non-trivial generalization of a super-linear lower bound for the size of Boolean formulas (treewidth 1) due to Neciporuk. Subsequently, we turn our attention to read-once circuits, which are circuits where each variable labels at most one input vertex. For each n, we show that any read-once circuit of treewidth t and size s computing a variant tau_n:{0,1}^n -> {0,1} of the element distinctness function must satisfy the inequality t * log(s) >= Omega(n/log(n)). Using this inequality in conjunction with known results in structural graph theory, we show that for each fixed graph H, read-once circuits computing tau_n which exclude H as a minor must have size at least Omega(n^2/log^{4}(n)). For certain well studied functions, such as the triangle-freeness function, this last lower bound can be improved to Omega(n^2/log^2(n)).
  • non-linear lower bounds
  • treewidth
  • element distinctness


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Michael Alekhnovich and Alexander A Razborov. Satisfiability, branch-width and Tseitin tautologies. In Proc. of the 43rd Symposium on Foundations of Computer Science, pages 593-603, 2002. Google Scholar
  2. Eric Allender, Shiteng Chen, Tiancheng Lou, Periklis A. Papakonstantinou, and Bangsheng Tang. Width-parametrized SAT: Time-space tradeoffs. Theory of Computing, 10(12):297-339, 2014. URL:
  3. Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 293-299. ACM, 1990. Google Scholar
  4. Norbert Blum. A boolean function requiring 3n network size. Theoretical Computer Science, 28(3):337-345, 1983. Google Scholar
  5. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. Google Scholar
  6. Elizabeth Broering and Satyanarayana V Lokam. Width-based algorithms for SAT and CIRCUIT-SAT. In Theory and Applications of Satisfiability Testing, pages 162-171. Springer, 2004. Google Scholar
  7. Chris Calabro. A lower bound on the size of series-parallel graphs dense in long paths. Electronic Colloquium on Computational Complexity (ECCC), 15(110), 2008. Google Scholar
  8. Evgeny Demenkov and Alexander S Kulikov. An elementary proof of a 3n-o(n) lower bound on the circuit complexity of affine dispersers. In Mathematical Foundations of Computer Science 2011, pages 256-265. Springer, 2011. Google Scholar
  9. Pavol Duris, Juraj Hromkovic, Stasys Jukna, Martin Sauerhoff, and Georg Schnitger. On multi-partition communication complexity. Information and Computation, 194(1):49-75, 2004. Google Scholar
  10. Magnus Gausdal Find, Alexander Golovnev, Edward A. Hirsch, and Alexander S. Kulikov. A better-than-3n lower bound for the circuit complexity of an explicit function. Electronic Colloquium on Computational Complexity (ECCC), 22(166), 2015. Google Scholar
  11. Anna Gál and Jing-Tang Jang. A generalization of spira’s theorem and circuits with small segregators or separators. In Theory and Practice of Computer Science (SOFSEM 2012), pages 264-276. Springer, 2012. Google Scholar
  12. Konstantinos Georgiou and Periklis A Papakonstantinou. Complexity and algorithms for well-structured k-sat instances. In Proc. of the 11th International Conference on Theory and Applications of Satisfiability Testing, pages 105-118. Springer, 2008. Google Scholar
  13. 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
  14. Martin Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica, 23(4):613-632, 2003. Google Scholar
  15. Johan Håstad. The shrinkage exponent of de morgan formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998. Google Scholar
  16. Jing He, Hongyu Liang, and Jayalal MN Sarma. Limiting negations in bounded treewidth and upward planar circuits. In Mathematical Foundations of Computer Science 2010, pages 417-428. Springer, 2010. Google Scholar
  17. Juraj Hromkovič. Communication complexity and lower bounds on multilective computations. RAIRO-Theoretical Informatics and Applications, 33(02):193-212, 1999. Google Scholar
  18. Kazuo Iwama and Hiroki Morizumi. An explicit lower bound of 5n- o (n) for boolean circuits. In Mathematical foundations of computer science 2002, pages 353-364. Springer, 2002. Google Scholar
  19. Maurice Jansen and Jayalal Sarma. Balancing bounded treewidth circuits. In Computer Science - Theory and Applications, pages 228-239. Springer, 2010. Google Scholar
  20. Stasys Jukna. Boolean function complexity: advances and frontiers, volume 27. Springer Science &Business Media, 2012. Google Scholar
  21. Oded Lachish and Ran Raz. Explicit lower bound of 4.5 n-o (n) for boolena circuits. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 399-408. ACM, 2001. Google Scholar
  22. Richard J Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM journal on computing, 9(3):615-627, 1980. Google Scholar
  23. Igor L Markov and Yaoyun Shi. Constant-degree graph expansions that preserve treewidth. Algorithmica, 59(4):461-470, 2011. Google Scholar
  24. Nečiporuk. On a Boolean function. Soviet Math. Dokl., 7(4):999-1000, 1966. Google Scholar
  25. Ramamohan Paturi and Pavel Pudlák. Circuit lower bounds and linear codes. Journal of Mathematical Sciences, 134(5):2425-2434, 2006. Google Scholar
  26. Pavel Pudlák. The hierarchy of boolean circuits. Computers and artificial intelligence, 6(5):449-468, 1987. Google Scholar
  27. Neil Robertson and Paul D Seymour. Graph minors. xiii. the disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. Google Scholar
  28. Rahul Santhanam and Srikanth Srinivasan. On the limits of sparsification. In Automata, Languages, and Programming, pages 774-785. Springer, 2012. Google Scholar
  29. John E. Savage. Planar circuit complexity and the performance of VLSI algorithms. In INRIA Report 77 (1981). Also in VLSI Systems and Computations, pages 61-67. Computer Science Press Rockwille MD, 1981. Google Scholar
  30. György Turán. On the complexity of planar boolean circuits. Computational Complexity, 5(1):24-42, 1995. Google Scholar
  31. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In 6th Symposium on Mathematical Foundations of Computer Science, pages 162-176, 1977. Google Scholar