Hitting sets for orbits of circuit classes and polynomial families

Saha, C and Thankey, B (2021) Hitting sets for orbits of circuit classes and polynomial families. In: 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021, 16-18 Aug 2021, Bengaluru; India.

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Abstract

The orbit of an n-variate polynomial f(x) over a field F is the set f(Ax+b): A � GL(n,F) and b � Fn. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: 1. Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials. 2. Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family IMM3,dd�N, which is complete for arithmetic formulas. 3. Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials. 4. Polynomials computable by occur-once formulas. © Chandan Saha and Bhargav Thankey; licensed under Creative Commons License CC-BY 4.0

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:	Computer circuits; Polynomial approximation, Arithmetic formulas; Depth 4; Elementary symmetric polynomial; Hitting sets; Multilinear polynomials; Polynomial-time; Power; Quasi-polynomial time; Rank concentration; ROABP, Timing circuits
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	29 Nov 2021 09:34
Last Modified:	29 Nov 2021 09:34
URI:	http://eprints.iisc.ac.in/id/eprint/70290

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