[Paper Review] A strong direct product theorem for two-way public coin communication complexity

Rahul Jain|arXiv (Cornell University)|Oct 5, 2010

Complexity and Algorithms in Graphs21 references3 citations

TL;DR

This paper establishes a strong direct product theorem for two-way public coin communication complexity by introducing a new complexity measure, the $ε$-error two-way conditional relative entropy bound ($\mathsf{crent}^2_\varepsilon(f)$), which generalizes the two-way product subdistribution bound. The authors prove that this measure provides tight lower bounds for the set disjointness problem, thereby reproducing Klauck's strong direct product result and unifying prior results under a single, tighter framework.

ABSTRACT

We show a direct product result for two-way public coin communication complexity of all relations in terms of a new complexity measure that we define. Our new measure is a generalization to non-product distributions of the two-way product subdistribution bound of [J, Klauck and Nayak 08], thereby our result implying their direct product result in terms of the two-way product subdistribution bound. We show that our new complexity measure gives tight lower bound for the set-disjointness problem, as a result we reproduce strong direct product result for this problem, which was previously shown by [Klauck 00].

Motivation & Objective

• To establish a strong direct product theorem for two-way public coin communication complexity.
• To define a new complexity measure that generalizes the two-way product subdistribution bound to non-product distributions.
• To demonstrate that the new measure yields tight lower bounds for the set disjointness problem.
• To unify and strengthen prior direct product results in communication complexity via a single, information-theoretic framework.

Proposed method

• Introduce the $\varepsilon$-error two-way conditional relative entropy bound ($\mathsf{crent}^2_\varepsilon(f)$) as a new complexity measure for relations.
• Prove that $\mathsf{crent}^2_\varepsilon(f)$ lower bounds the two-way public coin communication complexity $\mathsf{R}^{2,\mathsf{pub}}_\varepsilon(f)$.
• Show that $\mathsf{crent}^2_\varepsilon(f)$ upper bounds the two-way product subdistribution bound, thereby implying prior direct product results.
• Use the chain rule and joint convexity of relative entropy to analyze conditional distributions and protocol behavior across inputs.
• Apply the new measure to the set disjointness problem, showing it gives tight lower bounds.
• Leverage structural properties of the protocol and input distributions to bound success probability and derive the strong direct product result.

Experimental results

Research questions

• RQ1 Can a strong direct product theorem be proven for two-way public coin communication complexity using a generalized complexity measure?
• RQ2 Does the new $\mathsf{crent}^2_\varepsilon(f)$ measure provide a tighter or more general lower bound than existing measures like the two-way product subdistribution bound?
• RQ3 Can the new measure reproduce known strong direct product results, such as for the set disjointness problem?
• RQ4 Is the $\mathsf{crent}^2_\varepsilon(f)$ measure tight for the set disjointness problem?
• RQ5 How does the new measure unify or extend prior direct product results in communication complexity?

Key findings

• The new complexity measure $\mathsf{crent}^2_\varepsilon(f)$ forms a lower bound on the two-way public coin communication complexity $\mathsf{R}^{2,\mathsf{pub}}_\varepsilon(f)$.
• The measure $\mathsf{crent}^2_\varepsilon(f)$ generalizes the two-way product subdistribution bound, thereby implying the direct product result of J., Klauck, and Nayak [JKN08].
• For the set disjointness problem, $\mathsf{crent}^2_\varepsilon(f)$ gives a tight lower bound.
• The paper reproduces Klauck's strong direct product result for set disjointness using the new measure.
• The analysis shows that the success probability of computing $k$ copies of the relation in parallel is at most $(1 - \varepsilon)^k$ up to constant factors, confirming the strong direct product conjecture for this problem.
• The proof relies on bounding conditional probabilities and using properties of relative entropy and $\ell_1$ distance to control error propagation across inputs.