Document

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

An arithmetic formula is an arithmetic circuit where each gate has fan-out one. An arithmetic read-once formula (ROF in short) is an arithmetic formula where each input variable labels at most one leaf. In this paper we present several efficient blackbox polynomial identity testing (PIT) algorithms for some classes of polynomials related to read-once formulas. Namely, for polynomial of the form:
- f = Φ_1 ⋅ … ⋅ Φ_m + Ψ₁ ⋅ … ⋅ Ψ_r, where Φ_i,Ψ_j are ROFs for every i ∈ [m], j ∈ [r].
- f = Φ_1^{e₁} + Φ₂^{e₂} + Φ₃^{e₃}, where each Φ_i is an ROF and e_i-s are arbitrary positive integers.
Earlier, only a whitebox polynomial-time algorithm was known for the former class by Mahajan, Rao and Sreenivasaiah (Algorithmica 2016).
In the same paper, they also posed an open problem to come up with an efficient PIT algorithm for the class of polynomials of the form f = Φ_1^{e₁} + Φ_2^{e₂} + … + Φ_k^{e_k}, where each Φ_i is an ROF and k is some constant. Our second result answers this partially by giving a polynomial-time algorithm when k = 3. More generally, when each Φ₁,Φ₂,Φ₃ is a multilinear bounded-read formulae, we also give a quasi-polynomial-time blackbox PIT algorithm.
Our main technique relies on the hardness of representation approach introduced in Shpilka and Volkovich (Computational Complexity 2015). Specifically, we show hardness of representation for the resultant polynomial of two ROFs in our first result. For our second result, we lift hardness of representation for a sum of three ROFs to sum of their powers.

Pranav Bisht, Nikhil Gupta, and Ilya Volkovich. Towards Identity Testing for Sums of Products of Read-Once and Multilinear Bounded-Read Formulae. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2023.9, author = {Bisht, Pranav and Gupta, Nikhil and Volkovich, Ilya}, title = {{Towards Identity Testing for Sums of Products of Read-Once and Multilinear Bounded-Read Formulae}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {9:1--9:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.9}, URN = {urn:nbn:de:0030-drops-193829}, doi = {10.4230/LIPIcs.FSTTCS.2023.9}, annote = {Keywords: Identity Testing, Derandomization, Bounded-Read Formulae, Arithmetic Formulas} }

Document

**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP'10; Volkovich RANDOM'17; Bhargava, Saraf and Volkovich FOCS'18, JACM'20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring.
To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest.

Pranav Bisht and Nitin Saxena. Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2022.9, author = {Bisht, Pranav and Saxena, Nitin}, title = {{Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {9:1--9:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.9}, URN = {urn:nbn:de:0030-drops-174012}, doi = {10.4230/LIPIcs.FSTTCS.2022.9}, annote = {Keywords: Multivariate polynomial factorization, derandomization, sparse polynomials, symmetric polynomials, factor-sparsity, convex polytopes} }

Document

**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s^O(d²log n) terms. It is conjectured, though, that the "true" sparsity bound should be polynomial (i.e. s^poly(d)). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ^[2]ΠΣΠ^[ind-deg d] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.

Pranav Bisht and Ilya Volkovich. On Solving Sparse Polynomial Factorization Related Problems. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2022.10, author = {Bisht, Pranav and Volkovich, Ilya}, title = {{On Solving Sparse Polynomial Factorization Related Problems}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.10}, URN = {urn:nbn:de:0030-drops-174023}, doi = {10.4230/LIPIcs.FSTTCS.2022.10}, annote = {Keywords: Sparse Polynomials, Identity Testing, Derandomization, Factor-Sparsity, Multivariate Polynomial Factorization} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail