Refuting the Direct Sum Conjecture for Total Functions in Deterministic Communication Complexity
Simon Mackenzie, Abdallah Saffidine
TL;DR
The paper addresses whether solving multiple instances in deterministic two-party communication necessarily scales linearly with the number of instances. By constructing an alternating interlacing family of total functions and developing a projection/extraction framework, it proves a nontrivial lower bound that implies savings when instances are solved in batch. It then derives an upper bound for large instance counts and uses these bounds to refute the strongest form of the direct sum conjecture, showing that a constant-factor portion of total communication can be saved. The work introduces the overOperation and a subgame/projection toolkit that may extend to other direct-sum questions and related complexity models, with potential implications for circuit lower bounds and other distributed computation settings.
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 $(1 -ε)\ell$ times the complexity of solving one instance for some small enough $ε>0$. 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.