[Paper Review] A lower bound for essential covers of the cube

Gal Yehuda, Amir Yehudayoff|arXiv (Cornell University)|May 28, 2021

graph theory and CDMA systems9 references4 citations

TL;DR

This paper improves the lower bound for the minimum size of an essential cover of the n-dimensional hypercube from Ω(n⁰.⁵) to Ω(n⁰.⁵²), using structural analysis of hyperplane normal vectors and anti-concentration bounds via convex geometry. The result strengthens prior complexity-theoretic lower bounds in proof systems such as Res(linℚ) and stabbing planes proof systems.

ABSTRACT

Essential covers were introduced by Linial and Radhakrishnan as a model that captures two complementary properties: (1) all variables must be included and (2) no element is redundant. In their seminal paper, they proved that every essential cover of the $n$-dimensional hypercube must be of size at least $Ω(n^{0.5})$. Later on, this notion found several applications in complexity theory. We improve the lower bound to $Ω(n^{0.52})$, and describe two applications.

Motivation & Objective

• To establish a stronger lower bound on the minimum size of an essential cover of the n-dimensional hypercube, improving upon Linial and Radhakrishnan's Ω(n⁰.⁵) result.
• To address the challenge that essential covers are not generic, requiring new structural insights into the arrangement of hyperplane normal vectors.
• To apply the improved bound to strengthen existing lower bounds in proof complexity, particularly for resolution over linear equations and stabbing planes proof systems.
• To develop a novel method for locating uncovered vertices in a hypercube under essential cover constraints using iterative matrix reduction and anti-concentration arguments.

Proposed method

• Define an essential cover as a set of hyperplanes covering all vertices of the hypercube, with all variables appearing in at least one normal vector and no redundant hyperplanes.
• Use a structural lemma to partition the hyperplane matrix into row and column subsets, isolating sparse and structured components based on vector sparsity and scale decomposition.
• Apply an iterative reduction procedure on submatrices of the hyperplane matrix, maintaining conditions on sparsity and non-zero norms to identify critical structural components.
• Leverage anti-concentration bounds from convex geometry, particularly Theorem 15 from [13], to show that certain level sets (hyperplanes) cannot cover large antichains under product measures.
• Use a contradiction argument: assume a small cover exists, then show that a vertex must remain uncovered by analyzing column and row partitions and applying the anti-concentration bound.
• Employ a counting argument based on the sum of ℓ² norms across scale levels, using a power-law inequality (Claim 14) to derive a lower bound on the number of hyperplanes required.

Experimental results

Research questions

• RQ1 What is the best possible lower bound on the size of an essential cover of the n-dimensional hypercube?
• RQ2 How can structural constraints on hyperplane normal vectors be exploited to prove stronger lower bounds in non-generic settings?
• RQ3 Can improved lower bounds for essential covers be translated into stronger lower bounds in proof complexity systems?
• RQ4 To what extent can anti-concentration results for product measures be used to rule out small covers of the hypercube?

Key findings

• The paper establishes a new lower bound of Ω(n⁰.⁵²) for the minimum size of an essential cover of the n-dimensional hypercube, improving upon the previous Ω(n⁰.⁵) bound.
• The improved bound is derived via a novel structural analysis of the hyperplane matrix, partitioning rows and columns based on sparsity and multi-scale norms.
• The method identifies a vertex not covered by any hyperplane under the assumption of a small cover, using iterative matrix reduction and anti-concentration on antichains.
• The bound implies a stronger lower bound of 2^Ω(n⁰.⁵²) for tree-like Res(linℚ) proof systems, improving upon the prior 2^Ω(√n) bound.
• For the stabbing planes proof system, the bound yields a refutation-size lower bound of Ω(n¹.⁰⁴) for Tseitin formulas on the n×n grid, improving on the previous Ω(n) bound.
• The proof relies on a refined application of anti-concentration via Theorem 15, showing that level sets of linear forms under product measures have small measure unless the variance is small.