13 research outputs found
A simple and sharper proof of the hypergraph Moore bound
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., -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
, an appropriate generalization was conjectured by Feige [Fei08]. The
conjecture was settled up to an additional 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 , 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 factor), and loses only a single
logarithmic factor for all -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
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page