A simple and sharper proof of the hypergraph Moore bound

TL;DR

This paper presents a simpler, more concise proof of the hypergraph Moore bound, improving parameter tightness and extending to related problems in constraint satisfaction, compared to previous complex analyses.

Contribution

The authors introduce a new reweighted Kikuchi matrix and an edge deletion technique, simplifying the proof and tightening bounds for the hypergraph Moore bound and related problems.

Findings

Provides a shorter, simpler proof of the hypergraph Moore bound.

Achieves tighter parameters, matching classical bounds for graphs.

Extends methodology to refutation of smoothed constraint satisfaction problems.

Abstract

The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of hyperedges in a hypergraph. For graphs (i.e., $2$-uniform hypergraphs), a bound tight up to the leading constant was proven in a classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity $k>2$, an appropriate generalization was conjectured by Feige [Fei08]. The conjecture was settled up to an additional $\log^{4k+1} n$ factor in the size in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument relies on a connection between the existence of short even covers and the spectrum of a certain randomly signed Kikuchi matrix. Their analysis, especially for the case of odd $k$, is significantly complicated. In this work, we present a substantially simpler and shorter proof of the hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi matrix and an edge deletion step that allows us to drop several involved steps in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi matrix and the use of the Schudy-Sviridenko polynomial concentration. Our simpler proof also obtains tighter parameters: in particular, the argument gives a new proof of the classical Moore bound of [AHL02] with no loss (the proof in [GKM21] loses a $\log^3 n$ factor), and loses only a single logarithmic factor for all $k>2$-uniform hypergraphs. As in [GKM21], our ideas naturally extend to yield a simpler proof of the full trade-off for strongly refuting smoothed instances of constraint satisfaction problems with similarly improved parameters.