# A spectral bound on hypergraph discrepancy

## TL;DR

This paper establishes a spectral bound on hypergraph discrepancy using the maximum spectral value of a specific matrix, providing both theoretical insights and a polynomial-time coloring algorithm.

## Contribution

Introduces a spectral bound on hypergraph discrepancy based on the maximum spectral value, along with an efficient coloring algorithm.

## Key findings

- Discrepancy is bounded by $O(\sqrt{t} + \lambda)$ for $t$-regular hypergraphs.
- Random $t$-regular hypergraphs have discrepancy $O(\sqrt{t})$ with high probability.
- Provides a polynomial-time algorithm for hypergraph coloring with discrepancy guarantees.

## Abstract

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the discrepancy of $\mathcal{H}$ is $O(\sqrt{t} + \lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m \geq n$ edges is almost surely $O(\sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.