Worst-Case to Average-Case Reductions via Additive Combinatorics

TL;DR

This paper introduces a new framework leveraging additive combinatorics to transform algorithms with limited correctness into universally correct ones efficiently across various computational models.

Contribution

It provides explicit worst-case to average-case reductions for multiple fundamental problems using additive combinatorics techniques.

Findings

Efficient reductions for matrix multiplication algorithms.

Streaming algorithms improved for online matrix-vector multiplication.

Static data structures for linear problems and polynomial evaluation enhanced.

Abstract

We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time $T$ that are only correct on a small (subconstant) fraction of their inputs into algorithms running in time $\widetilde{O}(T)$ that are correct on all inputs. Using our framework, we obtain such efficient worst-case to average-case reductions for fundamental problems in a variety of computational models; namely, algorithms for matrix multiplication, streaming algorithms for the online matrix-vector multiplication problem, and static data structures for all linear problems as well as for the multivariate polynomial evaluation problem. Our techniques crucially rely on additive combinatorics. In particular, we show a local correction lemma that relies on a new probabilistic version of the quasi-polynomial Bogolyubov-Ruzsa lemma.