[Paper Review] Slicing the hypercube is not easy

Gal Yehuda, Amir Yehudayoff|arXiv (Cornell University)|Feb 10, 2021

Advanced Graph Theory Research23 references4 citations

TL;DR

This paper proves that slicing all edges of the $n$-dimensional hypercube requires at least $\Omega(n^{0.51})$ hyperplanes, improving the long-standing $\Omega(n^{0.5})$ lower bound by fifty years. The proof combines geometric insights, probabilistic methods, and anti-concentration inequalities to establish new lower bounds for threshold circuits and skew hyperplane coverings of the hypercube.

ABSTRACT

We prove that at least $Ω(n^{0.51})$ hyperplanes are needed to slice all edges of the $n$-dimensional hypercube. We provide a couple of applications: lower bounds on the computational complexity of parity, and a lower bound on the cover number of the hypercube by skew hyperplanes.

Motivation & Objective

• To establish a super-constant lower bound on the number of hyperplanes required to slice all edges of the $n$-dimensional hypercube.
• To improve upon the longstanding $\Omega(n^{0.5})$ lower bound for the slicing problem, which had not been improved in fifty years.
• To derive new lower bounds for the computational complexity of computing parity using threshold circuits.
• To provide a new lower bound on the number of skew hyperplanes needed to cover all vertices of the hypercube.
• To unify geometric, probabilistic, and combinatorial techniques in the analysis of hypercube slicing and covering problems.

Proposed method

• Uses a high-level connection to Tarski’s plank problem and Bang’s lemma to structure the slicing problem in terms of symmetric matrices and vector norms.
• Constructs a vector $w$ in the hypercube with controlled $\ell_\infty$-norm through iterative perturbations using a basis of vectors with small entries.
• Applies Bernstein’s inequality to bound the probability that a hyperplane with small coefficient norms slices a random edge, exploiting anti-concentration in product measures.
• Employs a two-phase construction: first handling hyperplanes with small $\ell_\infty$-norms via probabilistic concentration, then dealing with the remaining few via antichain arguments in product probability spaces.
• Uses a random perturbation $\delta$ on a base vector $w$ to generate candidate edges, ensuring the resulting edge distribution is a non-degenerate product measure.
• Applies Theorem 9 and union bounds over hyperplanes with large variance terms to show that the probability of slicing a random edge is negligible, implying a missing edge must exist under the assumed bound.

Experimental results

Research questions

• RQ1 Can the $\Omega(n^{0.5})$ lower bound on the number of hyperplanes needed to slice all edges of the $n$-cube be improved?
• RQ2 What is the minimum number of hyperplanes required to slice all edges of the $n$-dimensional hypercube, and does it exceed $n^{0.51}$?
• RQ3 What are the implications of this improved lower bound for the complexity of computing parity in threshold circuits?
• RQ4 What is the minimum number of skew hyperplanes required to cover all vertices of the $n$-dimensional hypercube?
• RQ5 Can probabilistic and anti-concentration techniques be used to prove stronger lower bounds in geometric and combinatorial problems involving the hypercube?

Key findings

• The paper establishes a new lower bound of $\Omega(n^{0.51})$ hyperplanes required to slice all edges of the $n$-dimensional hypercube, improving the previous $\Omega(n^{0.5})$ bound.
• This result implies the first $\omega(n^{0.5})$ lower bound on the number of gates in the first layer of any threshold circuit computing parity.
• It also yields the first $\omega(n^{1.5})$ lower bound on the number of wires in any depth-two threshold circuit for parity, specifically $\Omega(n^{1.51})$ wires.
• The same method implies a new $\Omega(n^{0.51})$ lower bound on the number of skew hyperplanes needed to cover all vertices of the hypercube.
• The analysis shows that for $k \leq n^{0.51}$ hyperplanes, there always exists at least one edge not sliced by any hyperplane, via a probabilistic construction and anti-concentration arguments.
• The proof combines geometric vector construction, Bernstein’s inequality for sub-Gaussian tails, and antichain analysis in product probability spaces to achieve the improved bound.