NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Baraskar, O and Dewan, A and Saha, C and Sinha, P (2024) NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In: 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, 8 July 2024through 12 July 2024, Tallinn.

PDF lei_int_pro_inf_297_2024 - Published Version Restricted to Registered users only Download (883kB) | Request a copy

Abstract

An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f � Fx and s � N, ETsparse asks to check if there exist A � GL(|x|, F) and b � F|x| such that f(Ax + b) is s-sparse. We show that ETsparse is NP-hard over any field F, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s0 such that a given s-sparse polynomial f is in the orbit of some 1 s0-sparse polynomial to within a factor of s3 �ϵ is NP-hard for any ϵ > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant � � 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-� polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem. © Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Equivalence classes, 3SAT; Equivalence testing; Equivalence to; Minimum circuit size problem; Non-zero coefficients; NP-hard; NP-hardness; Sparse polynomials; Sparse representation; Testing equivalence, Polynomial approximation
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	18 Dec 2024 04:54
Last Modified:	18 Dec 2024 04:54
URI:	http://eprints.iisc.ac.in/id/eprint/85833

Actions (login required)

View Item