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Communication Lower Bounds via Critical Block Sensitivity

We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (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 Nordström: 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 $Ω(b)$ 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 $Ω(n/\log n)$; previously, a bound of $Ω(\sqrt{n})$ was known (Raz and Wigderson, JACM 1992). Moreover, we prove a $Θ(\sqrt{n})$ 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 Lovász--Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.