[Paper Review] On the Sensitivity Conjecture for Disjunctive Normal Forms

Chengyu Lin, Shengyu Zhang|arXiv (Cornell University)|Jan 1, 2016

Complexity and Algorithms in Graphs54 references15 citations

TL;DR

This paper establishes new upper bounds on block sensitivity and communication complexity for Boolean functions with small alternating numbers, extending the Sensitivity and Log-rank Conjectures beyond monotone functions. It proves that block sensitivity is polynomially bounded by the square of the alternating number times sensitivity, and communication complexity for XOR and AND composed functions is bounded by polynomial functions of the alternating number and log-rank, respectively.

ABSTRACT

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.

Motivation & Objective

• To extend the Sensitivity Conjecture and Log-rank Conjecture beyond monotone functions to functions with small alternating numbers.
• To analyze how the complexity measures—block sensitivity, communication complexity, and degree—behave under small alternations from monotonicity.
• To establish polynomial upper bounds on block sensitivity and communication complexity in terms of alternating number and log-rank.
• To generalize prior results on monotone functions to a broader class of functions that are close to monotone in structure.

Proposed method

• Define the alternating number as the maximum number of value changes along any monotone path from 0^n to 1^n.
• Use induction and recursive bounding techniques on the 1-covering number of communication matrices for AND and XOR composed functions.
• Apply bounds based on max terms (for 1-inputs), min terms (for 0-inputs), and communication complexity itself to derive recursive inequalities.
• Leverage known connections between communication matrix rank and sparsity: M"obius sparsity for AND functions and Fourier sparsity for XOR functions.
• Use the fact that subfunctions restricted to cubes with fixed support have reduced alternating number, enabling inductive bounds.
• Combine inequalities from three types of bounds (max terms, min terms, and direct CC bounds) to derive tight overall communication complexity bounds.

Experimental results

Research questions

• RQ1 Does the Sensitivity Conjecture hold for functions with polynomially bounded alternating number relative to sensitivity?
• RQ2 Can the Log-rank Conjecture be extended to XOR and AND composed functions when the outer function has small alternating number?
• RQ3 How does the communication complexity of f◦⊕ and f◦∧ scale with the alternating number of f and the log-rank of the communication matrix?

Key findings

• Block sensitivity bs(f) is bounded by O(alt(f)^2 · s(f)), confirming the Sensitivity Conjecture for functions with alt(f) = poly(s(f)).
• Communication complexity for XOR functions satisfies CC(f◦⊕) ≤ 2 · alt(f) · log₂(rank(Mf◦⊕)), proving the Log-rank Conjecture for f with alt(f) = poly(log(∥f̂∥₀)).
• For AND functions, CC(f◦∧) ≤ O(log^{alt(f)+1}(rank(Mf◦∧))), establishing the Log-rank Conjecture for f with bounded alternating number.
• A refined bound improves the dependence on alt(f) to O(log^{alt(f)/2 + 1}(rank(Mf◦∧)) · log n) when tolerating a log n factor.
• The results generalize prior results on monotone functions to a broader class of functions with small alternating number.
• The analysis shows that small alternating number implies strong structural control over communication complexity and sensitivity, even beyond monotonicity.