Refuting the Direct Sum Conjecture for Total Functions in Deterministic Communication Complexity

TL;DR

This paper presents a counterexample to the direct sum conjecture in deterministic communication complexity, showing that solving multiple instances can require less combined resources than solving each separately.

Contribution

It introduces a family of total functions demonstrating that the complexity of multiple instances can be less than the sum of individual complexities, refuting the conjecture.

Findings

Counterexample functions show less than linear scaling in complexity

Method forces round alternation to exploit complexity measures

Demonstrates a gap between joint and separate instance complexities

Abstract

In communication complexity the input of a function $f:X\times Y\rightarrow Z$ is distributed between two players Alice and Bob. If Alice knows only $x\in X$ and Bob only $y\in Y$, how much information must Alice and Bob share to be able to elicit the value of $f(x,y)$? Do we need $\ell$ more resources to solve $\ell$ instances of a problem? This question is the direct sum question and has been studied in many computational models. In this paper we focus on the case of 2-party deterministic communication complexity and give a counterexample to the direct sum conjecture in its strongest form. To do so we exhibit a family of functions for which the complexity of solving $\ell$ instances is less than $(1 -\epsilon )\ell$ times the complexity of solving one instance for some small enough $\epsilon>0$. We use a customised method in the analysis of our family of total functions, showing that one can force the alternation of rounds between players. This idea allows us to exploit the integrality of the complexity measure to create an increasing gap between the complexity of solving the instances independently with that of solving them together.