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Sparsity Lower Bounds for Probabilistic Polynomials

Authors Josh Alman , Arkadev Chattopadhyay , Ryan Williams

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Author Details

Josh Alman
  • Columbia University, New York, NY, USA
Arkadev Chattopadhyay
  • TIFR Mumbai, India
Ryan Williams
  • CSAIL and EECS, MIT, Cambridge, MA, USA

Funding

  • Alman, Josh: Work supported in part by NSF Grant CCF-2238221.
  • Chattopadhyay, Arkadev: Supported by funds of Department of Atomic Energy, Govt. of India, under project RTI4001, and a Google India Research Award.
  • Williams, Ryan: Work supported in part by NSF CCF-2127597 and NSF CCF-2420092.

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Josh Alman, Arkadev Chattopadhyay, and Ryan Williams. Sparsity Lower Bounds for Probabilistic Polynomials. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 3:1-3:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.3

Abstract

Probabilistic polynomials over commutative rings offer a powerful way of representing Boolean functions. Although many degree lower bounds for such representations have been proved, sparsity lower bounds (counting the number of monomials in the polynomials) have not been so common. Sparsity upper bounds are of great interest for potential algorithmic applications, since sparse probabilistic polynomials are the key technical tool behind the best known algorithms for many core problems, including dense All-Pairs Shortest Paths, and the existence of sparser polynomials would lead to breakthrough algorithms for these problems.
In this paper, we prove several strong lower bounds on the sparsity of probabilistic and approximate polynomials computing Boolean functions when 0 means "false". Our main result is that the AND of n ORs of c log n variables requires probabilistic polynomials (over any commutative ring which isn't too large) of sparsity n^Ω(log c) to achieve even 1/4 error. The lower bound is tight, and it rules out a large class of polynomial-method approaches for refuting the APSP and SETH conjectures via matrix multiplication. Our other results include:  
- Every probabilistic polynomial (over a commutative ring) for the disjointness function on two n-bit vectors requires exponential sparsity in order to achieve exponentially low error. 
- A generic lower bound that any function requiring probabilistic polynomials of degree d must require probabilistic polynomials of sparsity Ω(2^d). 
- Building on earlier work, we consider the probabilistic rank of Boolean functions which generalizes the notion of sparsity for probabilistic polynomials, and prove separations of probabilistic rank and probabilistic sparsity. 
Some of our results and lemmas are basis independent. For example, over any basis {a,b} for true and false where a ≠ b, and any commutative ring R, the AND function on n variables has no probabilistic R-polynomial with 2^o(n) sparsity, o(n) degree, and 1/2^o(n) error simultaneously. This AND lower bound is our main technical lemma used in the above lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Models of computation
  • Mathematics of computing → Discrete mathematics
Keywords
  • Probabilistic Polynomials
  • Sparsity
  • Orthogonal Vectors
  • Probabilistic Rank

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