[Paper Review] An $O(n^{0.4732})$ upper bound on the complexity of the GKS communication game

Márió Szegedy|arXiv (Cornell University)|Jun 22, 2015

Complexity and Algorithms in Graphs2 references3 citations

TL;DR

This paper presents an $O(n^{0.4732})$ upper bound on the communication complexity of the GKS game, improving upon the previous $\sqrt{999/1000}\sqrt{n}$ bound. The result is achieved by constructing a $(5,30)$ strategy and applying a recursive composition lemma, which enables a superpolynomial improvement in the exponent of $n$ and brings the complexity closer to resolving the Sensitivity Conjecture.

ABSTRACT

We give an $5\cdot n^{\log_{30}5}$ upper bund on the complexity of the communication game introduced by G. Gilmer, M. Koucký and M. Saks \cite{saks} to study the Sensitivity Conjecture \cite{linial}, improving on their $\sqrt{999\over 1000}\sqrt{n}$ bound. We also determine the exact complexity of the game up to $n\le 9$.

Motivation & Objective

• To improve the upper bound on the communication complexity of the GKS game, which is central to the Sensitivity Conjecture.
• To construct a more efficient strategy for the GKS game by designing a $(5,30)$ strategy with bounded output size.
• To leverage recursive composition of strategies to derive a sub-polynomial upper bound on $k(n)$, the minimal size of Bob's guess set.
• To determine the exact value of $k(n)$ for $n \leq 9$, providing a baseline for future analysis.

Proposed method

• Constructing a $(5,30)$ strategy by partitioning the $n=30$ elements into 5 blocks of size 6 and applying a consistent assignment rule based on partial permutation history.
• Using a recursive composition lemma: if $(k,n)$ and $(k',n')$ strategies exist, then a $(kk',nn')$ strategy exists, enabling exponent reduction through repeated composition.
• Defining Alice’s strategy to maintain parity constraints across blocks so that Bob can reconstruct Merlin’s move via block-wise parity checks.
• Defining Bob’s strategy to compute block-wise parity sums and apply the outer strategy to narrow down the possible positions of Merlin’s move to a small set.
• Proving that no $(2,5)$ strategy exists by showing that the required disjoint 8-cycles for each coordinate would exceed the total number of nodes.
• Using structural analysis of path and cycle shapes in the graph $G$ to eliminate all configurations except the unique 8-cycle of Type 7, which supports the $(5,30)$ strategy.

Experimental results

Research questions

• RQ1 Can the communication complexity of the GKS game be improved beyond the $\sqrt{999/1000}\sqrt{n}$ bound?
• RQ2 What is the minimal $k(n)$ such that a $(k,n)$ strategy exists, and how does it scale with $n$?
• RQ3 Is there a $(5,30)$ strategy that enables recursive composition to yield a sub-polynomial complexity?
• RQ4 Can the exact value of $k(n)$ be determined for small $n$, particularly $n \leq 9$?
• RQ5 Does the existence of a $(5,30)$ strategy imply a non-trivial upper bound on $k(n)$ that improves the exponent of $n$?

Key findings

• The paper establishes an $O(n^{0.4732})$ upper bound on the complexity of the GKS communication game, significantly improving over the previous $\sqrt{999/1000}\sqrt{n}$ bound.
• A $(5,30)$ strategy is explicitly constructed, which serves as a base case for recursive composition and enables the improved exponent.
• The recursive composition lemma allows combining strategies to achieve $k(nn') \leq k(n)k(n')$, leading to the $n^{\log_{30}5}$ scaling, where $\log_{30}5 \approx 0.4732$.
• The exact value of $k(n)$ is determined for $n \leq 9$, with $k(5) = 3$, $k(6) = 3$, and $k(9) = 3$, showing that $k(n)$ grows slowly.
• It is proven that no $(2,5)$ strategy exists, which rules out a simpler construction and supports the optimality of the current approach.
• The only possible connected component of the graph $G$ corresponding to the strategy is the unique 8-cycle of Type 7, which confirms the structural consistency of the $(5,30)$ construction.