Lower bounds on the DNF exception problem for short exception lists and related problems

TL;DR

This paper establishes new lower bounds on the complexity of the DNF exception problem for short exception lists, using linear programming relaxations, with improved bounds for small exception lists relative to previous results.

Contribution

It introduces a linear programming based method to derive stronger lower bounds for the DNF exception problem with short exception lists, advancing theoretical understanding.

Findings

New explicit lower bounds for exception list size up to logarithm of dimension

Bounds are significantly stronger than previous known bounds

Method applicable to hypercube covering problems

Abstract

In this paper we prowide lower bounds on the complexity of the DNF exception problem for short exception lists and hypercube covering problem. The method proposed is based on the relaxation of the initial problem to a certain linear programming problem. Some explicit bounds are provided for the case when exception list size is bounded above by a logarithm of dimension. The bound provided in this case is significantly stronger than the bounds known before.