Communication Lower Bounds via Critical Block Sensitivity [article]

Mika Göös, Toniann Pitassi
2016 arXiv   pre-print
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
more » ... 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/ n); previously, a bound of Ω(√(n)) was known (Raz and Wigderson, JACM 1992). Moreover, we prove a Θ(√(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.
arXiv:1311.2355v2 fatcat:xhrth42o2nbqbaytaol2e34gt4