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Improved Explicit Hitting-Sets for ROABPs

Zeyu Guo, Rohit Gurjar, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We give improved explicit constructions of hitting-sets for read-once oblivious algebraic branching programs (ROABPs) and related models.  ...  Finally, we give improved explicit hitting-sets for polynomials computable by width-r ROABPs in any variable order, also known as any-order ROABPs.  ...  In this work we give improved explicit hitting-sets for ROABPs (unknown order), sum of several ROABPs (small variate regime) and any-order ROABPs with respect to various parameters.  ... 
doi:10.4230/lipics.approx/random.2020.4 dblp:conf/approx/GuoG20 fatcat:kpw3hzowtbhinh7ln4mpmnpr3y

Hitting sets for multilinear read-once algebraic branching programs, in any order

Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14  
We note that the model of multilinear ROABPs contains the model of set-multilinear algebraic branching programs, which itself contains the model of set-multilinear formulas of arbitrary depth.  ...  We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n O(lg 2 n) time. 1 Further, our algorithm is oblivious  ...  The first two authors would like to thank Chandan Saha for explaining the work of Agrawal-Saha-Saxena [ASS12] to them.  ... 
doi:10.1145/2591796.2591816 dblp:conf/stoc/ForbesSS14 fatcat:ipjeduquyvhc3ndiistq2qt6se

Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order [article]

Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka
2013 arXiv   pre-print
We note that the model of multilinear ROABPs contains the model of set-multilinear algebraic branching programs, which itself contains the model of set-multilinear formulas of arbitrary depth.  ...  We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time.  ...  The first two authors would like to thank Chandan Saha for explaining the work of Agrawal-Saha-Saxena [ASS12] to them.  ... 
arXiv:1309.5668v1 fatcat:jbjxqwhlujhjtlzpn55zntbuni

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing [article]

Michael A. Forbes, Amir Shpilka
2013 arXiv   pre-print
Previous work have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich.  ...  That is, we improve Mulmuley's reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization.  ...  He would also like to thank Sergey Yekhanin for the conversation that led to Theorem A.1, and Scott Aaronson for some helpful comments.  ... 
arXiv:1303.0084v2 fatcat:mvpzlhkjnnelri4s6nezomojoe

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing [chapter]

Michael A. Forbes, Amir Shpilka
2013 Lecture Notes in Computer Science  
Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model.  ...  That is, we improve Mulmuley's reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization.  ...  Let H ⊆ F n 2 be a t(n, r)-explicit hitting set for width ≤ 2n 2 , depth n 2 , degree < r ROABPs.  ... 
doi:10.1007/978-3-642-40328-6_37 fatcat:jd4yqz6a7jgdfewml4inacgsvm

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)).  ...  In the blackbox setting, hitting sets of quasipolynomial size were obtained in [24, 22, 2] , where the last two papers being applicable even if the order in which the variable are read is unknown (the  ...  Theorem 2. 1 ( 1 Hitting Set for ROABPs, [2]).  ... 
doi:10.1145/3170709 fatcat:fnoqrflnq5dzlgyvcpavkq3en4

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

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2015 arXiv   pre-print
We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm.  ...  Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs).  ...  Forbes, Shpilka and Saptharishi [FSS14] obtained a hitting set of size (nwd) O(d log(w) log n) for unknown order ROABPs. This was improved later by Agrawal et al.  ... 
arXiv:1511.07136v1 fatcat:ndotfilu25h2zji75r4iajbkmq

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

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 arXiv   pre-print
Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices.  ...  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.  ...  We also thank the anonymous reviewers for their careful reading of this paper and for many useful comments.  ... 
arXiv:1701.05328v2 fatcat:ueg2mlf5l5febo7hojpqri67dq

Deterministic Divisibility Testing via Shifted Partial Derivatives

Michael A. Forbes
2015 2015 IEEE 56th Annual Symposium on Foundations of Computer Science  
We give explicit sums of powers of quadratic polynomials that require exponentially-large roABPs in a strong sense, showing that techniques known for roABPs have limited applicability in our regime.  ...  Creating deterministic PIT algorithms is a significant challenge, as it is known to have implications for the existence of explicit polynomials that require large algebraic circuits for their computation  ...  We would also like to thank Amir Shpilka in particular for the question of how to deterministically test whether a quadratic polynomial divides a sparse polynomial, Chandan Saha for questions that led  ... 
doi:10.1109/focs.2015.35 dblp:conf/focs/Forbes15 fatcat:zkoa4aj23fgnvcm7lsswvun7i4

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  
Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices.  ...  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.  ...  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

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, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little  ...  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

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

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

Progress on Polynomial Identity Testing - II [article]

Nitin Saxena
2014 arXiv   pre-print
Using a different technique [AGKS13] also proves constantconcentration, hence designs poly-time hitting-sets, for certain constant-width ROABP.  ...  Rank concentration, shift, hitting-sets The hitting-sets that we saw till now were for models where some parameter was kept bounded.  ... 
arXiv:1401.0976v1 fatcat:wrc3gfl2ajbmbhl2reb6lm722m

Pseudorandom Bits for Oblivious Branching Programs [article]

Rohit Gurjar, Ben Lee Volk
2017 arXiv   pre-print
For polynomial width branching programs, the seed lengths in our constructions are Õ(n^1-1/2^k-1) (for the read-k case) and O(n/ n) (for the linear length case).  ...  Previously, the best construction for these models required seed length (1-Ω(1))n.  ...  Acknowledgment We thank Andrej Bogdanov for useful comments on an earlier version of this text.  ... 
arXiv:1708.02054v1 fatcat:na6jizwj7bdqxebyoievmor7i4
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