A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints

TL;DR

This paper advances the understanding of formula size lower bounds by introducing a new LP-based technique using clique constraints, improving bounds for majority functions and recursive ternary majority functions.

Contribution

It develops a novel LP bound method leveraging the stable set polytope, providing improved lower bounds for specific monotone functions and matching upper bounds for recursive ternary majority functions.

Findings

Improved formula size lower bounds for majority functions.

Established matching upper and lower bounds for unbalanced recursive ternary majority functions.

Enhanced lower bounds for balanced recursive ternary majority functions using quantum adversary bounds.

Abstract

We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan [11] and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko [13]. Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy [15].