Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits
There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n).
We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.
Balancing sets
depth-2 threshold circuits
polynomials
majority
weighted thresholds
Mathematics of computing~Combinatorics
Theory of computation~Circuit complexity
72:1-72:14
Track A: Algorithms, Complexity and Games
This work was done while the authors were visiting the Simons Institute for the Theory of Computing.
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/026/.
Pavel
Hrubeš
Pavel Hrubeš
Institute of Mathematics of ASCR, Prague
Supported by ERC grant FEALORA 339691 and the GACR grant 19-27871X.
Sivaramakrishnan
Natarajan Ramamoorthy
Sivaramakrishnan Natarajan Ramamoorthy
Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Supported by the National Science Foundation under agreement CCF- 1420268.
Anup
Rao
Anup Rao
Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Supported by the National Science Foundation under agreement CCF- 1420268.
Amir
Yehudayoff
Amir Yehudayoff
Department of Mathematics, Technion-IIT, Haifa, Israel
Partially supported by ISF grant 1162/15.
10.4230/LIPIcs.ICALP.2019.72
M. Ajtai, J. Komlós, and E. Szemerédi. Sorting in c logn parallel steps. Combinatorica, 3(1):1-19, March 1983. URL: http://dx.doi.org/10.1007/BF02579338.
http://dx.doi.org/10.1007/BF02579338
Eric Allender and Michal Koucký. Amplifying lower bounds by means of self-reducibility. Journal of the ACM, 57(3):1-36, March 2010. URL: http://dx.doi.org/10.1145/1706591.1706594.
http://dx.doi.org/10.1145/1706591.1706594
Noga Alon. Personal Communication, 2019.
Noga Alon, Ernest E. Bergmann, Don Coppersmith, and Andrew M. Odlyzko. Balancing sets of vectors. IEEE Transactions on Information Theory, 34(1):128-130, 1988.
Noga Alon, Mrinal Kumar, and Ben Lee Volk. Unbalancing sets and an almost quadratic lower bound for syntactically multilinear arithmetic circuits. arXiv, 2017. URL: http://arxiv.org/abs/1708.02037.
http://arxiv.org/abs/1708.02037
Kazuyuki Amano. Depth Two Majority Circuits for Majority and List Expanders. In Igor Potapov, Paul Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), volume 117, pages 81:1-81:13, 2018. URL: http://drops.dagstuhl.de/opus/volltexte/2018/9663.
http://drops.dagstuhl.de/opus/volltexte/2018/9663
Kazuyuki Amano and Masafumi Yoshida. Depth Two (n-2)-Majority Circuits for n-Majority. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E101.A(9):1543-1545, 2018.
Roger C. Baker, Glyn Harman, and János Pintz. The difference between consecutive primes, II. Proceedings of the London Mathematical Society, 83(3):532-562, 2001.
Christian Engels, Mohit Garg, Kazuhisa Makino, and Anup Rao. On expressing majority as a majority of majorities. In Electronic Colloquium on Computational Complexity (ECCC), volume 24, page 174, 2017.
Hikoe Enomoto, Peter Frankl, Noboru Ito, and Kazumasa Nomura. Codes with given distances. Graphs and Combinatorics, 3(1):25-38, 1987.
David Eppstein and Daniel S. Hirschberg. From discrepancy to majority. Algorithmica, 80(4):1278-1297, 2018.
Peter Frankl and Vojtěch Rödl. Forbidden intersections. Transactions of the American Mathematical Society, 300(1):259-286, 1987.
Kristoffer Arnsfelt Hansen and Vladimir V. Podolskii. Exact threshold circuits. In 2010 25th Annual IEEE Conference on Computational Complexity, pages 270-279. IEEE, 2010.
Johan Håstad, Guillaume Lagarde, and Joseph Swernofsky. d-Galvin families. arXiv, January 2019. URL: http://arxiv.org/abs/1901.02652.
http://arxiv.org/abs/1901.02652
Gábor Hegedűs. Balancing sets of vectors. Studia Scientiarum Mathematicarum Hungarica, 47(3):333-349, 2009.
Maurice J. Jansen. Lower bounds for syntactically multilinear algebraic branching programs. In International Symposium on Mathematical Foundations of Computer Science, pages 407-418, 2008.
Alexander S. Kulikov and Vladimir V. Podolskii. Computing majority by constant depth majority circuits with low fan-in gates. arXiv, 2016. URL: http://arxiv.org/abs/1610.02686.
http://arxiv.org/abs/1610.02686
L. Lovász. Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25(3):319-324, 1978.
Gleb Posobin. Computing majority with low-fan-in majority queries. arXiv, 2017. URL: http://arxiv.org/abs/1711.10176.
http://arxiv.org/abs/1711.10176
Ran Raz, Amir Shpilka, and Amir Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM Journal on Computing, 38(4):1624-1647, 2008.
Srikanth Srinivasan. Personal Communication, 2018.
L.G. Valiant. Short monotone formulae for the majority function. Journal of Algorithms, 5(3):363-366, 1984. URL: http://dx.doi.org/10.1016/0196-6774(84)90016-6.
http://dx.doi.org/10.1016/0196-6774(84)90016-6
R. Ryan Williams. Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials. arXiv, 2018. URL: http://arxiv.org/abs/1802.09121.
http://arxiv.org/abs/1802.09121
Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode