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. • Monotone circuit depth: We exhibit a monotone function on n variables whose monotone circuits require depth Ω(n/log n); previously, a bound of Ω(√n was known (Raz and Wigderson, JACM 1992). Moreover, we prove a tight Θ(√n) monotone depth bound for a function in monotone P. This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013). • 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.