[Paper Review] Discrepancy in random hypergraph models

Aditya Potukuchi|arXiv (Cornell University)|Nov 5, 2018

Mathematical Approximation and Integration13 references9 citations

TL;DR

This paper studies hypergraph discrepancy in two random models: $χ_1$ (t-regular hypergraphs with each vertex in exactly $t$ edges) and $χ_2$ (i.i.d. incidence matrix with edge probability $p$). For $χ_1$, it proves that when $\log^{10}n \ll t \ll \sqrt{n}$ and $m=n$, the discrepancy is almost surely $O(\sqrt{t})$, improving prior bounds. For $χ_2$, when $p=1/2$ and $n=\Omega(m\log m)$, the discrepancy is almost surely at most 1, resolving an open problem of Hoberg and Rothvoss.

ABSTRACT

We study hypergraph discrepancy in two closely related random models of hypergraphs on $n$ vertices and $m$ hyperedges. The first model, $\mathcal{H}_1$, is when every vertex is present in exactly $t$ randomly chosen hyperedges. The premise of this is closely tied to, and motivated by the Beck-Fiala conjecture. The second, perhaps more natural model, $\mathcal{H}_2$, is when the entries of the $m imes n$ incidence matrix is sampled in an i.i.d. fashion, each with probability $p$. We prove the following: 1. In $\mathcal{H}_1$, when $\log^{10}n \ll t \ll \sqrt{n}$, and $m = n$, we show that the discrepancy of the hypergraph is almost surely at most $O(\sqrt{t})$. This improves upon a result of Ezra and Lovett for this range of parameters. 2. In $\mathcal{H}_2$, when $p= \frac{1}{2}$, and $n = Ω(m \log m)$, we show that the discrepancy is almost surely at most $1$. This answers an open problem of Hoberg and Rothvoss.

Motivation & Objective

• To analyze the discrepancy of random hypergraphs in two closely related models: $\mathcal{H}_1$ (t-regular) and $\mathcal{H}_2$ (i.i.d. incidence).
• To improve upon existing upper bounds for hypergraph discrepancy in the $\mathcal{H}_1$ model for large $t$.
• To resolve an open problem by Hoberg and Rothvoss on the discrepancy of $\mathcal{H}_2$ when $p=1/2$ and $n=\Omega(m\log m)$.
• To provide a constructive algorithm for achieving the $O(\sqrt{t})$ discrepancy bound in $\mathcal{H}_1$.

Proposed method

• Uses the partial coloring method inspired by Spencer and Lovett-Meka to iteratively color vertices and reduce discrepancy.
• Applies a martingale concentration argument with bounded differences to control the deviation of linear forms over random incidence matrices.
• Employs a bounded variance martingale inequality to show that the sum of signed entries over a subset of the incidence matrix concentrates around its mean.
• Introduces a structured decomposition of the incidence matrix into low-impact and high-impact parts to isolate and bound large deviations.
• Uses the Lovász Local Lemma framework with modifications to handle large hyperedges in $\mathcal{H}_1$, ensuring their discrepancy remains small.
• Establishes a constructive algorithm via the partial coloring lemma, with runtime in randomized $\text{poly}(n)$ time.

Experimental results

Research questions

• RQ1 What is the typical discrepancy of a random $t$-regular hypergraph when $t$ is large but $t \ll \sqrt{n}$, and $m=n$?
• RQ2 Can the $O(\sqrt{t})$ discrepancy bound from the Beck-Fiala conjecture be achieved in random $t$-regular hypergraphs for this parameter range?
• RQ3 What is the discrepancy of a random hypergraph with i.i.d. $\text{Bernoulli}(1/2)$ incidence entries when $n=\Omega(m\log m)$?
• RQ4 Is it possible to achieve discrepancy 1 in such a model, and does this resolve the open problem posed by Hoberg and Rothvoss?
• RQ5 Can the constructive partial coloring method be adapted to yield efficient algorithms for discrepancy minimization in random hypergraph models?

Key findings

• For $\mathcal{H}_1$, when $C_1\log^{10}n \ll t \ll \sqrt{n}$ and $m=n$, the discrepancy is almost surely at most $O(\sqrt{t})$, improving on Ezra and Lovett's $O(\sqrt{t\log t})$ bound.
• The $O(\sqrt{t})$ discrepancy bound is achieved via a constructive algorithm running in randomized $\text{poly}(n)$ time.
• For $\mathcal{H}_2$, when $p=1/2$ and $n=\Omega(m\log m)$, the discrepancy is almost surely at most 1, resolving an open problem of Hoberg and Rothvoss.
• The proof relies on a martingale concentration argument with bounded differences and variance, showing that the sum of signed entries over a structured subset of the incidence matrix concentrates tightly.
• The analysis shows that the expected contribution of the random matrix entries to the discrepancy is bounded by $O(\sqrt{t})$, and high-probability concentration follows from bounded variance martingale inequalities.
• The result for $\mathcal{H}_2$ implies that random hypergraphs with $p=1/2$ and sufficiently many vertices relative to edges have near-optimal discrepancy of 1 with high probability.