A LOCAL View of the Polynomial Hierarchy

Abstract

We extend classical methods of computational complexity to the realm of distributed computing, where they sometimes prove more effective than in their original context. Our focus is on decision problems in the LOCAL model, a setting in which networked computers use synchronous message passing to collectively answer questions about their network topology. We impose two time constraints on this model: the number of communication rounds is bounded by a constant, and the number of computation steps of each computer is polynomially bounded in the size of its local input and received messages. By letting two players alternately assign certificates to all computers, we obtain a distributed generalization of the polynomial hierarchy (and thus of the complexity classes $\mathbf{P}$ and $\mathbf{NP}$). We then extend key results of complexity theory to this setting, including the Cook-Levin theorem (which identifies Boolean satisfiability as a complete problem for $\mathbf{NP}$) and Fagin's theorem (which characterizes $\mathbf{NP}$ as the class of problems expressible in existential second-order logic). The original results can be recovered as the special case where the network consists of a single computer. But perhaps more surprisingly, separating complexity classes becomes easier in the distributed setting: we can show that our hierarchy is infinite, while it remains notoriously open whether the same holds when restricted to a single computer. (By contrast, a collapse of our hierarchy would have implied a collapse of the classical polynomial hierarchy.) As an application, we propose quantifier alternation as a new tool for measuring the locality of problems in distributed computing.