Lower bounds on information complexity via zero-communication protocols and applications

TL;DR

This paper establishes that most known lower bound methods for communication complexity also serve as lower bounds for information complexity, introducing a relaxed partition bound and a connection to zero-communication protocols to derive new lower bounds.

Contribution

It introduces a relaxed partition bound that lower bounds information complexity and connects zero-communication protocols to lower bounds, resolving open questions in the field.

Findings

Lower bounds for information complexity via relaxed partition bound

Exponential separation between quantum and classical information complexity for certain problems

Omega(n) lower bound for the Gap Hamming Distance problem

Abstract

We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm based methods (e.g. the factorization norm method) and rectangle-based methods (e.g. the rectangle/corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols where the players can either output a value or abort. We prove the following compression lemma: given a protocol for a function f with information complexity I, one can construct a zero-communication protocol that has non-abort probability at least 2^{-O(I)} and that computes f correctly with high probability conditioned on not aborting. Then, we show how such a zero-communication protocol relates to the relaxed partition bound. We use our main theorem to resolve three of the open questions raised by Braverman. First, we show that the information complexity of the Vector in Subspace Problem is {\Omega}(n^{1/3}), which, in turn, implies that there exists an exponential separation between quantum communication complexity and classical information complexity. Moreover, we provide an {\Omega}(n) lower bound on the information complexity of the Gap Hamming Distance Problem.