We prove a lower bound of Ω(n2/log2n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x1,...,xn). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff (), who proved a lower bound of Ω(n4/3/log2n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.
A special case of our combinatorial result implies, for every n, a tight Ω(n) lower bound on the minimum size of a family F of subsets of cardinality 2n of a set X of size 4n, so that any subset of X of size 2n has intersection of size exactly n with some member of F. This settles a problem of Galvin up to a constant factor, extending results of Frankl and Rödl  and Enomoto et al. , who proved in 1987 the above statement (with a tight constant) for odd values of n, leaving the even case open.
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N. Alon, E. E. Bergmann, D. Coppersmith and A. M. Odlyzko: Balancing sets of vectors, IEEE Trans. Information Theory34 (1988), 128–130.
R. P. Anstee, L. Rónyai and A. Sali: Shattering news, Graphs and Combinatorics18 (2002), 59–73.
N. Alon and J. H. Spencer: The Probabilistic Method, Wiley Publishing, 4th edition, 2016.
S. J. Berkowitz: On computing the determinant in small parallel time using a small number of processors, Information Processing Letters18 (1984), 147–150.
R. C. Baker, G. Harman and J. Pintz: The difference between consecutive primes, ii, Proceedings of the London Mathematical Society83 (2001), 532–562.
W. Baur and V. Strassen: The complexity of partial derivatives, Theoretical Computer Science22 (1983), 317–330.
S. Chillara, C. Engels, N. Limaye and S. Srinivasan: A near-optimal depth-hierarchy theorem for small-depth multilinear circuits, in: Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018), pages 934–945. IEEE Computer Society, 2018.
X. Chen, N. Kayal and A. Wigderson: Partial Derivatives in Arithmetic Complexity (and beyond), Foundation and Trends in Theoretical Computer Science, 2011.
S. Chillara, N. Limaye and S. Srinivasan: A quadratic size-hierarchy theorem for small-depth multilinear formulas, in: Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP 2018), volume 107 of LIPIcs, 36:1-36:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
S. Chillara, N. Limaye and S. Srinivasan: Small-depth multilinear formula lower bounds for iterated matrix multiplication, with applications, in: Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), volume 96 of LIPIcs, 21:1-21:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
L. Csanky: Fast parallel matrix inversion algorithms, SIAM J. Comput.5 (1976), 618–623.
H. Enomoto, P. Frankl, N. Ito and K. Nomura: Codes with given distances, Graphs Combin. 3 (1987), 25–38.
J. B. Farr and S. Gao: Computing gröbner bases for vanishing ideals of finite sets of points, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 16th International Symposium, AAECC-16, Las Vegas, NV, USA, February 20–24, 2006, Proceedings, 118–127, 2006.
H. Fournier, N. Limaye, G. Malod and S. Srinivasan: Lower bounds for depth 4 formulas computing iterated matrix multiplication, in: Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), 128–135, 2014.
P. Frankl and V. Rödl: Forbidden intersections, Trans. Amer. Math. Soc.300 (1987), 259–286.
D. Grigoriev and M. Karpinski: An exponential lower bound for depth 3 arithmetic circuits, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC 1998), 577–582, 1998.
A. Gupta, P. Kamath, N. Kayal and R. Saptharishi: Approaching the chasm at depth four, Journal of the ACM61 (2014), 33:1-33:16.
D. Grigoriev and A. A. Razborov: Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields, Appl. Algebra Eng. Commun. Comput.10 (2000), 465–487
G. Hegedűs: Balancing sets of vectors, Studia Sci. Math. Hungar.47 (2010), 333–349.
G. Hegedűs and L. Rónyai: Gröbner bases for complete uniform families, J. Algebraic Combin.17 (2003), 171–180.
M. J. Jansen: Lower bounds for syntactically multilinear algebraic branching programs, in: Proceedings of the 33rd Internationl Symposium on the Mathematical Foundations of Computer Science (MFCS 2008), volume 5162 of Lecture Notes in Computer Science, pages 407–418. Springer, 2008.
K. Kalorkoti: A Lower Bound for the Formula Size of Rational Functions, SIAM J. Comput.14 (1985), 678–687.
N. Kayal, N. Limaye, C. Saha and S. Srinivasan: An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Circuits, in: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), 61–70, 2014.
D. E. Knuth: Efficient balanced codes, IEEE Trans. Information Theory32 (1986), 51–53.
M. Kumar and S. Saraf: On the power of homogeneous depth 4 arithmetic circuits, in: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), 364–373, 2014.
M. Kumar and R. Saptharishi: An exponential lower bound for homogeneous depth-5 circuits over finite fields, in: Proceedings of the 32nd Annual Computational Complexity Conference (CCC 2017), volume 79, 31:1-31:30, 2017.
M. Kumar: A quadratic lower bound for homogeneous algebraic branching programs, in: Proceedings of the 32nd Annual Computational Complexity Conference (CCC 2017), volume 79, 19:1-19:16, 2017.
M. Mahajan and V. Vinay: Determinant: Combinatorics, algorithms, and complexity, Chicago J. Theor. Comput. Sci., 1997.
N. Nisan: Lower bounds for non-commutative computation, in: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC 1991), 410–418, 1991.
N. Nisan and A. Wigderson: Lower bounds on arithmetic circuits via partial derivatives, Computational Complexity6 (1997), 217–234
R. Raz: Separation of multilinear circuit and formula size, Theory of Computing2 (2006), 121–135
R. Raz: Multi-linear formulas for permanent and determinant are of super-polynomial size, J. ACM56 (2009), 8:1-8:17.
R. Raz: Elusive functions and lower bounds for arithmetic circuits, Theory of Computing6 (2010), 135–177.
R. Raz, A. Shpilka and A. Yehudayoff: A lower bound for the size of syntactically multilinear arithmetic circuits, SIAM J. Comput.38 (2008), 1624–1647
R. Raz and A. Yehudayoff: Balancing syntactically multilinear arithmetic circuits, Computational Complexity17 (2008), 515–535.
R. Raz and A. Yehudayoff: Lower bounds and separations for constant depth multilinear circuits, Computational Complexity18 (2009), 171–207
R. Saptharishi: A survey of lower bounds in arithmetic circuit complexity, Github survey, https://github.com/dasarpmar/lowerbounds-survey/, 2016.
M. Skala: Hypergeometric tail inequalities: ending the insanity, arXiv preprint arXiv:1311.5939, 2013.
V. Strassen: Die berechnungskomplexität von elementarsymmetrischen funktionen und von interpolationskoeffizienten, Numerische Mathematik20 (1973), 238–251.
A. Shpilka and A. Yehudayoff: Arithmetic circuits: A survey of recent results and open questions, Foundations and Trends in Theoretical Computer Science5 (2010), 207–388.
Part of this work was done while the second author was visiting Tel Aviv University. We thank Amir Shpilka for the visit, for many insightful discussions, and for comments on an earlier version of this text. We also thank Srikanth Srivinasan for allowing us to include his proof of Lemma 2.1 in this paper, and Andy Drucker and the anonymous reviewers for their suggestions and comments on earlier versions of the manuscript.
Part of this work was done at the Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, USA.
Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the Simons Foundation.
Part of this work was done at the School of Computer Science, Tel Aviv University, Tel Aviv, Israel. The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16).
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Alon, N., Kumar, M. & Volk, B.L. Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits. Combinatorica 40, 149–178 (2020). https://doi.org/10.1007/s00493-019-4009-0
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