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Improved Explicit Hitting-Sets for ROABPs
2020
International Workshop on Approximation Algorithms for Combinatorial Optimization
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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
2014
Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
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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]
2013
arXiv
pre-print
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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]
2013
arXiv
pre-print
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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]
2013
Lecture Notes in Computer Science
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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
2018
ACM Transactions on Computation Theory
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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]
2015
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2015
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
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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
2017
Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017
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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
2018
Theory of Computing
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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
2016
Computational Complexity
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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]
2014
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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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