The Polynomial Method in Circuit Complexity Applied to Algorithm Design (Invited Talk)

Author Richard Ryan Williams

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Richard Ryan Williams

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Richard Ryan Williams. The Polynomial Method in Circuit Complexity Applied to Algorithm Design (Invited Talk). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 47-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


In circuit complexity, the polynomial method is a general approach to proving circuit lower bounds in restricted settings. One shows that functions computed by sufficiently restricted circuits are "correlated" in some way with a low-complexity polynomial, where complexity may be measured by the degree of the polynomial or the number of monomials. Then, results limiting the capabilities of low-complexity polynomials are extended to the restricted circuits. Old theorems proved by this method have recently found interesting applications to the design of algorithms for basic problems in the theory of computing. This paper surveys some of these applications, and gives a few new ones.
  • algorithm design
  • circuit complexity
  • polynomial method


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  1. Scott Aaronson. The polynomial method in quantum and classical computing. In FOCS, pages 3-3. IEEE, 2008. Google Scholar
  2. Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In SODA, 2015. Google Scholar
  3. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In FOCS, 2014. Google Scholar
  4. Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In SODA, page to appear, 2015. Google Scholar
  5. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  6. James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich. The expressive power of voting polynomials. Combinatorica, 14(2):135-148, 1994. Google Scholar
  7. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald De Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001. Google Scholar
  8. Richard Beigel. The polynomial method in circuit complexity. In In Proceedings of the 8th IEEE Structure in Complexity Theory Conference, pages 82-95. IEEE Computer Society Press, 1995. Google Scholar
  9. Richard Beigel, Nick Reingold, and Daniel Spielman. The perceptron strikes back. In Structure in Complexity Theory Conference, pages 286-291. IEEE, 1991. Google Scholar
  10. Richard Beigel and Jun Tarui. On ACC. Computational Complexity, pages 350-366, 1994. Google Scholar
  11. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. A duality between clause width and clause density for SAT. In IEEE Conf. Computational Complexity, pages 252-260, 2006. Google Scholar
  12. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In Parameterized and Exact Computation, pages 75-85. Springer, 2009. Google Scholar
  13. Timothy M. Chan. All-pairs shortest paths with real weights in O(n³/log n) time. Algorithmica, 50(2):236-243, 2008. See also WADS'05. Google Scholar
  14. Timothy M. Chan. More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput., 39(5):2075-2089, 2010. See also STOC'07. Google Scholar
  15. Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM J. Comput., 11(3):467-471, 1982. Google Scholar
  16. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. Google Scholar
  17. Robert W. Floyd. Algorithm 97. Comm. ACM, 5-6:345, 1962. Google Scholar
  18. Michael L. Fredman. New bounds on the complexity of the shortest path problem. SIAM J. Comput., 5(1):49-60, 1976. See also FOCS'75. Google Scholar
  19. Parikshit Gopalan and Rocco A. Servedio. Learning and lower bounds for AC0 with threshold gates. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 588-601. Springer, 2010. Google Scholar
  20. Yijie Han. Improved algorithm for all pairs shortest paths. Information Processing Letters, 91(5):245-250, 2004. Google Scholar
  21. Yijie Han. An O(n³ (log log n / log n)^5/4) time algorithm for all pairs shortest path. Algorithmica, 51(4):428-434, 2008. See also ESA'06. Google Scholar
  22. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. Google Scholar
  23. Richard M. Karp. Dynamic programming meets the principle of inclusion and exclusion. Operations Research Letters, 1(2):49-51, 1982. Google Scholar
  24. Adam R. Klivans and Rocco Servedio. Learning DNF in time 2^O(n^1/3). In STOC, pages 258-265. ACM, 2001. Google Scholar
  25. Swastik Kopparty and Srikanth Srinivasan. Certifying polynomials for AC⁰(parity) circuits, with applications. In FSTTCS, pages 36-47, 2012. Google Scholar
  26. Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. J. ACM, 40(3):607-620, 1993. Google Scholar
  27. Yishay Mansour. An o(n^log log n) learning algorithm for dnf under the uniform distribution. Journal of Computer and System Sciences, 50(3):543-550, 1995. Google Scholar
  28. Marvin Minsky and Seymour Papert. Perceptrons. MIT Press, 1969. Google Scholar
  29. Elchanan Mossel, Ryan O'Donnell, and Rocco A. Servedio. Learning functions of k relevant variables. Journal of Computer and System Sciences, 69(3):421-434, 2004. Google Scholar
  30. Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms. Cambridge University Press, 1995. Google Scholar
  31. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Computational Complexity, 4(4):301-313, 1994. Google Scholar
  32. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. Google Scholar
  33. A. A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987. Google Scholar
  34. Liam Roditty and Uri Zwick. On dynamic shortest paths problems. In Algorithms-ESA 2004, pages 580-591. Springer, 2004. Google Scholar
  35. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. Google Scholar
  36. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In STOC, pages 77-82, 1987. Google Scholar
  37. Tadao Takaoka. A new upper bound on the complexity of the all pairs shortest path problem. Information Processing Letters, 43(4):195-199, 1992. See also WG'91. Google Scholar
  38. Tadao Takaoka. Subcubic cost algorithms for the all pairs shortest path problem. Algorithmica, 20(3):309-318, 1998. See also WG'95. Google Scholar
  39. Tadao Takaoka. An O(n³ log log n/log n) time algorithm for the all-pairs shortest path problem. Information Processing Letters, 96(5):155-161, 2005. Google Scholar
  40. Jun Tarui. Probabilistic polynomials, AC0 functions and the polynomial-time hierarchy. Theoretical computer science, 113(1):167-183, 1993. Google Scholar
  41. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix and triangle problems. In FOCS, pages 645-654. IEEE, 2010. Google Scholar
  42. Heribert Vollmer. Introduction to circuit complexity: a uniform approach. Springer, 1999. Google Scholar
  43. Stephen Warshall. A theorem on Boolean matrices. J. ACM, 9:11-12, 1962. Google Scholar
  44. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. See also ICALP'04. Google Scholar
  45. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In STOC, pages 664-673, 2014. Google Scholar
  46. Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. In STOC, pages 194-202, 2014. Google Scholar
  47. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2, 2014. Google Scholar
  48. Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In SODA, pages 1867-1877. SIAM, 2014. Google Scholar
  49. Andrew Chi-Chih Yao. On ACC and threshold circuits. In FOCS, pages 619-627, 1990. Google Scholar
  50. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Lecture Notes in Computer Science, volume 72, pages 216-226. Springer, 1979. Google Scholar
  51. Uri Zwick. A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths. In ISAAC 2004, volume 3341 of Springer LNCS, pages 921-932, 2004. Google Scholar
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