A simple and sharper proof of the hypergraph Moore bound

07/22/2022
by   Jun-Ting Hsieh, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/09/2022

Spectral hypergraph sparsification via chaining

In a hypergraph on n vertices where D is the maximum size of a hyperedge...
research
08/17/2020

Approximate Hypergraph Vertex Cover and generalized Tuza's conjecture

A famous conjecture of Tuza states that the minimum number of edges need...
research
06/20/2023

Online Vector Bin Packing and Hypergraph Coloring Illuminated: Simpler Proofs and New Connections

This paper studies the online vector bin packing (OVBP) problem and the ...
research
08/24/2022

A Simpler Proof that Pairing Heaps Take O(1) Amortized Time per Insertion

The pairing heap is a simple "self-adjusting" implementation of a heap (...
research
02/15/2019

Universally Sparse Hypergraphs with Applications to Coding Theory

For fixed integers r> 2,e> 2,v> r+1, an r-uniform hypergraph is called G...
research
05/29/2022

A note on hardness of promise hypergraph colouring

We show a slightly simpler proof the following theorem by I. Dinur, O. R...
research
08/11/2023

Simple Analysis of Priority Sampling

We prove a tight upper bound on the variance of the priority sampling me...

Please sign up or login with your details

Forgot password? Click here to reset