The communication complexity of XOR games via summing operators

TL;DR

This paper demonstrates that the discrepancy method can be arbitrarily ineffective for certain XOR games by applying p-summing operator theory from Banach space analysis to establish new bounds and insights.

Contribution

It introduces a novel application of p-summing operators to show the discrepancy method's limitations and constructs XOR games with prescribed communication complexity bounds.

Findings

Discrepancy method can be arbitrarily poor for some XOR games.

Existence of XOR games with any desired value of the communication complexity.

Application of Banach space theory to communication complexity analysis.

Abstract

The discrepancy method is widely used to find lower bounds for communication complexity of XOR games. It is well known that these bounds can be far from optimal. In this context Disjointness is usually mentioned as a case where the method fails to give good bounds, because the increment of the value of the game is linear (rather than exponential) in the number of communicated bits. We show in this paper the existence of XOR games where the discrepancy method yields bounds as poor as one desires. Indeed, we show the existence of such games with any previously prescribed value. To prove this result we apply the theory of p-summing operators, a central topic in Banach space theory. We show in the paper other applications of this theory to the study of the communication complexity of XOR games.