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Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas [article]

Rafael Oliveira, Amir Shpilka, Ben Lee Volk
2014 arXiv   pre-print
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas.  ...  For depth-3 multilinear formulas, of size (n^δ), we give a hitting set of size (Õ(n^2/3 + 2δ/3)).  ...  Acknowledgments The authors would like to thank Zeev Dvir and Avi Wigderson for helpful discussions during the course of this work.  ... 
arXiv:1411.7492v1 fatcat:tktkstrumbbhpf6lvymrq6mtda

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Rafael Oliveira, Amir Shpilka, Ben lee Volk
2016 Computational Complexity  
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas.  ...  For depth-3 multilinear formulas, of size exp(n δ ), we give a hitting set of size exp Õ n 2/3+2δ/3 .  ...  The authors would like to thank Zeev Dvir and Avi Wigderson for helpful discussions during the course of this work.  ... 
doi:10.1007/s00037-016-0131-1 fatcat:m55ugwfj6jbm5gqqirfxszw5le

Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in

Zohar S. Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for  ...  More precisely, he shows that the black-box derandomization of PIT implies that an explicit multilinear polynomial has no subexponential size arithmetic circuit.  ...  The best algorithms today are for sums of read-once formulas [SV09] and for set-multilinear depth 3 formulas (non black-box) [RS05] .  ... 
doi:10.1145/1806689.1806779 dblp:conf/stoc/KarninMSV10 fatcat:sklwzqypifbuhmw6snvzaanpka

Deterministic Identity Testing of Depth-4 Multilinear Circuits with Bounded Top Fan-in

Zohar S. Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich
2013 SIAM journal on computing (Print)  
Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for  ...  More precisely, he shows that the black-box derandomization of PIT implies that an explicit multilinear polynomial has no subexponential size arithmetic circuit.  ...  The best algorithms today are for sums of read-once formulas [SV09] and for set-multilinear depth 3 formulas (non black-box) [RS05] .  ... 
doi:10.1137/110824516 fatcat:7kwyeftqrbbcho5vpw4hu6syze

Proving Lower Bounds Via Pseudo-random Generators [chapter]

Manindra Agrawal
2005 Lecture Notes in Computer Science  
In this paper, we formalize two stepwise approaches, based on pseudo-random generators, for proving P = NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits.  ...  This implies that f B,d cannot be a hitting set generator for all d (because SAC 1 circuits can be transformed to subexponential sized constant depth circuits as observed earlier).  ...  We conjecture that above lemma holds for all depths: Conjecture. Function f A,d is a hitting set generator against depth d, size n arithmetic circuits for every d > 0.  ... 
doi:10.1007/11590156_6 fatcat:utnfe6pyvfczxkyaqpy6gchrdu

Deterministic polynomial identity tests for multilinear bounded-read formulae

Matthew Anderson, Dieter van Melkebeek, Ilya Volkovich
2015 Computational Complexity  
Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four formulae.  ...  Before our work no subexponential-time deterministic algorithm was known for this class of formulae.  ...  Acknowledgements The authors would like to thank Amir Shpilka for bringing them in touch with each other, and the anonymous reviewers for their comments.  ... 
doi:10.1007/s00037-015-0097-4 fatcat:5z7zqjuifjetnltve7qeugx5se

Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds [article]

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 arXiv   pre-print
That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits.  ...  Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.  ...  We also thank the anonymous reviewers for their careful reading of this paper and for many useful comments.  ... 
arXiv:1701.05328v2 fatcat:ueg2mlf5l5febo7hojpqri67dq

Succinct hitting sets and barriers to proving algebraic circuits lower bounds

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits.  ...  Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.  ...  Acknowledgements We thank Scott Aaronson, Andy Drucker, Josh Grochow, Mrinal Kumar, Shubhangi Saraf and Dor Minzer for useful conversations regarding this work.  ... 
doi:10.1145/3055399.3055496 dblp:conf/stoc/ForbesSV17 fatcat:amglevgewvdlpozsxppzhdqify

Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2018 ACM Transactions on Computation Theory  
hitting of [22] was quasipolynomial sized for bounded individual degree, but the subsequent hitting set of [2] is quasipolynomial sized for any d = poly(n)).  ...  -4 circuits, that maps subexponential size to subexponential size [3] .  ...  Shpilka and Volkovich [61] constructed quasipolynomial-size hitting set for read-once formulas, and Anderson, van Melkebeek and Volkovich [8] extended this result to multilinear read-k formulas and  ... 
doi:10.1145/3170709 fatcat:fnoqrflnq5dzlgyvcpavkq3en4

Quasi-polynomial hitting-set for set-depth-Δ formulas

Manindra Agrawal, Chandan Saha, Nitin Saxena
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
In this work, we give a hitting-set generator for set-depth-∆ formulas (over any field) with running time polynomial in exp((∆ 2 log s) ∆−1 ), where s is the size bound on the input set-depth-∆ formula  ...  with bounded top fanin [ASSS12, SS11], • depth-4 (bounded depth) constant-occur formulas [ASSS12], and a quasi-polynomial time hitting-set generator formultilinear constant-read formulas [AvMV11],  ...  Acknowledgments This work was initiated when MA and NS visited Max Planck Institute for Informatics, and would like to thank the institute for its generous hospitality.  ... 
doi:10.1145/2488608.2488649 dblp:conf/stoc/AgrawalSS13 fatcat:5ajt6l7etneghgq2kqxyssukmu

Quasi-polynomial Hitting-set for Set-depth-Delta Formulas [article]

Manindra Agrawal, Chandan Saha, Nitin Saxena
2012 arXiv   pre-print
In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp((Delta^2 log s)^Delta - 1), where s is the size bound on the input set-depth-Delta  ...  Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator.  ...  Acknowledgments This work was initiated when MA and NS visited Max Planck Institute for Informatics, and would like to thank the institute for its generous hospitality.  ... 
arXiv:1209.2333v1 fatcat:52f2hvrvfbgh5g35c5yic7jsei

Hardness-Randomness Tradeoffs for Algebraic Computation

Mrinal Kumar, Ramprasad Saptharishi
2019 Bulletin of the European Association for Theoretical Computer Science  
The interplay between the question of proving lower bounds and that of derandomization, in various settings, is one of the central themes in complexity theory.  ...  We also talk about an application of this machinery to the phenomenon of bootstrapping for polynomial identity testing and mention some open problems.  ...  Acknowledgements We are thankful to Chi-Ning Chou, Zeyu Guo, Swastik Kopparty, Noam Solomon, Anamay Tengse and Ben Lee Volk for insightful discussions on many of the themes addressed in this survey.  ... 
dblp:journals/eatcs/0001S19 fatcat:kwvkfr7ghrd47l4ytdord7jaii

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 Theory of Computing  
polynomial identity testing problem, that is, to the existence of a hitting set for the class of poly(N)-degree poly(N)-size circuits which consists of coefficient vectors of polynomials of polylog(N)  ...  However, Razborov and Rudich [86] showed that (possibly after a small modification) most circuit lower bounds (such as those for constant-depth circuits ([9, 41, 106, 53, 85, 98])) yield large and constructive  ...  We also thank the anonymous reviewers for their careful reading of this paper and for many useful comments.  ... 
doi:10.4086/toc.2018.v014a018 dblp:journals/toc/ForbesSV18 fatcat:xca443ndhzfxfjnclcmx6y57py

Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

Manindra Agrawal, Rohit Gurjar, Arpita Korwar, Nitin Saxena
2015 SIAM journal on computing (Print)  
The 1-distance strictly subsumes the set-multilinear model, while n-distance captures general multilinear depth-3. We design a hitting-set in time (nk) O(∆ log n) for ∆-distance.  ...  Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper, we take a step towards designing such hitting-sets.  ...  On the other hand, there are exponential lower bounds for depth-3 multilinear circuits [RY09] .  ... 
doi:10.1137/140975103 fatcat:bcs2sbyu2nchrlzzympgl7eowu

Hitting-sets for ROABP and Sum of Set-Multilinear circuits [article]

Manindra Agrawal and Rohit Gurjar and Arpita Korwar and Nitin Saxena
2014 arXiv   pre-print
We give the first subexponential whitebox PIT for the sum of constantly many set-multilinear depth-3 circuits. To achieve this, we define notions of distance and base sets.  ...  Distance, for a multilinear depth-3 circuit, measures how far are the partitions from a mere refinement. We design a hitting-set in time n^O(d n) for d-distance.  ...  On the other hand, there are exponential lower bounds for depth-3 multilinear circuits [RY09] .  ... 
arXiv:1406.7535v1 fatcat:qlqaxcqvkzasnpd7rdd62azbba
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