Optimal Hitting Sets for Combinatorial Shapes

TL;DR

This paper presents an optimal construction of explicit hitting sets for Combinatorial Shapes, generalizing previous results and improving the size from quasipolynomial to polynomial in key parameters, with potential independent applications.

Contribution

It introduces a polynomial-sized explicit hitting set for Combinatorial Shapes, extending prior work and providing stronger guarantees for related combinatorial structures.

Findings

Constructed polynomial-sized hitting sets for Combinatorial Shapes.

Extended previous results to more general classes of tests.

Developed fractional perfect hash families with stronger guarantees.

Abstract

We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests, including symmetric functions and combinatorial rectangles. Generalizing results of Linial, Luby, Saks, and Zuckerman (Combinatorica 1997) and Rabani and Shpilka (SICOMP 2010), we construct hitting sets for Combinatorial Shapes of size polynomial in the alphabet, dimension, and the inverse of the error parameter. This is optimal up to polynomial factors. The best previous hitting sets came from the Pseudorandom Generator construction of Gopalan et al., and in particular had size that was quasipolynomial in the inverse of the error parameter. Our construction builds on natural variants of the constructions of Linial et al. and Rabani and Shpilka. In the process, we construct fractional perfect hash families and hitting sets for combinatorial rectangles with stronger guarantees. These might be of independent interest.