[Paper Review] A Composition Theorem for Conical Juntas

Jain, Rahul, Lee, Troy+1 more|arXiv (Cornell University)|Jan 18, 2014

Quantum Computing Algorithms and Architecture1 references11 citations

TL;DR

This paper introduces the public-coin partition bound, a new linear programming-based lower bound method for classical communication and query complexity. It proves that the logarithm of this bound is quadratically tight: it lower bounds public-coin randomized communication and query complexity, and its square upper bounds the complexity, making it the first known bound to achieve this tightness for all relations.

ABSTRACT

In this work we introduce, both for classical communication complexity and query complexity, a modification of the 'partition bound' introduced by Jain and Klauck [2010]. We call it the 'public-coin partition bound'. We show that (the logarithm to the base two of) its communication complexity and query complexity versions form, for all relations, a quadratically tight lower bound on the public-coin randomized communication complexity and randomized query complexity respectively.

Motivation & Objective

• To address the open question of how tight the partition bound is for communication and query complexity.
• To develop a new variant of the partition bound that incorporates public-coin randomness, improving its strength and applicability.
• To establish that the new bound is quadratically tight, meaning its logarithm is both a lower bound and its square is an upper bound on public-coin randomized complexity.
• To unify and strengthen existing lower bound techniques in communication and query complexity using linear programming.

Proposed method

• Define the public-coin partition bound as a linear program that models partitions of input spaces into rectangles or assignments, each associated with an output and a probability.
• In the primal, variables wz,R represent the probability of outputting z on input (x,y) within rectangle R, constrained to sum to 1 over all rectangles per input.
• Introduce a probability distribution aP over partitions P, where wz,R is the sum of aP over all partitions P containing (z,R), ensuring consistency with the protocol's output distribution.
• Use the dual form of the linear program to derive a maximization problem that bounds the complexity via weights µx,y, φx,y, and auxiliary variables vz,R.
• Prove that the logarithm of the optimal value of the primal is a lower bound on public-coin randomized communication and query complexity.
• Show that the square of the logarithm of the optimal value upper bounds the complexity, by constructing a protocol that samples partitions and simulates them with bounded communication or query cost.

Experimental results

Research questions

• RQ1 Is the partition bound quadratically tight for public-coin randomized communication complexity?
• RQ2 Can a modified partition bound be constructed that captures public-coin randomness and achieves tightness?
• RQ3 Does the new bound strictly dominate the original partition bound in strength for all relations?
• RQ4 Can the new bound be used to establish tight upper and lower bounds on communication and query complexity simultaneously?

Key findings

• The logarithm of the public-coin partition bound is a lower bound on public-coin randomized communication complexity for any relation.
• The square of the logarithm of the public-coin partition bound upper bounds public-coin randomized communication complexity, establishing quadratic tightness.
• The public-coin partition bound is strictly stronger than the original partition bound for all relations in both communication and query complexity.
• For query complexity, the logarithm of the bound lower bounds randomized query complexity, and its square upper bounds the complexity.
• The bound is quadratically tight: for any relation, the public-coin complexity is sandwiched between log₂(pprtε(f)) and (log₂(pprtε(f)) + log₂(1/ε) + 1)².
• The new bound is defined via a linear program that explicitly models public-coin protocols as distributions over partitions, enabling tighter analysis.