Separating decision tree complexity from subcube partition complexity

TL;DR

This paper demonstrates a separation between subcube partition complexity and randomized decision tree complexity, introduces new lower bounds, and shows the limitations of existing techniques in proving optimal bounds.

Contribution

It establishes a separation between subcube partition and decision tree complexities and analyzes the effectiveness of lower bound methods.

Findings

Existence of a function with smaller subcube partition complexity than decision tree complexity.

Public-coin partition bound also lower bounds subcube partition complexity.

Current lower bound techniques cannot prove optimal bounds for randomized decision trees.

Abstract

The subcube partition model of computation is at least as powerful as decision trees but no separation between these models was known. We show that there exists a function whose deterministic subcube partition complexity is asymptotically smaller than its randomized decision tree complexity, resolving an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is based on the information-theoretic techniques first introduced to lower bound the randomized decision tree complexity of the recursive majority function. We also show that the public-coin partition bound, the best known lower bound method for randomized decision tree complexity subsuming other general techniques such as block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound, also lower bounds randomized subcube partition complexity. This shows that all these lower bound techniques cannot prove optimal lower bounds for randomized decision tree complexity, which answers an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014).