Communication Lower Bounds via Critical Block Sensitivity

TL;DR

This paper introduces a new proof technique using critical block sensitivity to establish communication lower bounds, leading to advances in monotone circuit depth and proof complexity for semi-algebraic systems.

Contribution

It provides a simplified, generalizable proof of communication lower bounds via critical block sensitivity and applies it to derive new results in circuit depth and proof complexity.

Findings

Monotone circuit depth for NP functions is at least Ω(n/log n)

First length-space lower bounds for semi-algebraic proof systems

Simplified proof technique generalizes to multi-party communication complexity

Abstract

We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordstr\"om (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordstr\"om: if $S$ is a search problem with critical block sensitivity $b$, then every randomised two-party protocol solving a certain two-party lift of $S$ requires $\Omega(b)$ bits of communication. Besides simplicity, our proof has the advantage of generalising to the multi-party setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications: (1) Monotone Circuit Depth: We exhibit a monotone $n$-variable function in NP whose monotone circuits require depth $\Omega(n/\log n)$; previously, a bound of $\Omega(\sqrt{n})$ was known (Raz and Wigderson, JACM 1992). Moreover, we prove a $\Theta(\sqrt{n})$ monotone depth bound for a function in monotone P. (2) Proof Complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lov\'asz--Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordstr\"om.