Decision list compression by mild random restrictions

TL;DR

This paper proves that decision lists with small width can be approximated by smaller decision lists, resolving a conjecture on DNF sparsification, using a novel random restriction lemma that fixes only a small fraction of variables.

Contribution

It introduces a new random restriction lemma and proves decision list approximation bounds, resolving a key conjecture in complexity theory.

Findings

Small-width decision lists can be approximated by small-size decision lists.

A new random restriction lemma allows analysis with minimal variable fixing.

Resolved a conjecture on DNF sparsification from 2013.

Abstract

A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs, and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed.