9 research outputs found
An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
We prove the following Alon-Boppana type theorem for general (not necessarily
regular) weighted graphs: if is an -node weighted undirected graph of
average combinatorial degree (that is, has edges) and girth , and if are the
eigenvalues of the (non-normalized) Laplacian of , then (The Alon-Boppana theorem implies that if is unweighted and
-regular, then if the diameter is at least .)
Our result implies a lower bound for spectral sparsifiers. A graph is a
spectral -sparsifier of a graph if where is the Laplacian matrix of and is
the Laplacian matrix of . Batson, Spielman and Srivastava proved that for
every there is an -sparsifier of average degree where
and the edges of are a
(weighted) subset of the edges of . Batson, Spielman and Srivastava also
show that the bound on cannot be reduced below when is a clique; our Alon-Boppana-type result implies that
cannot be reduced below when comes
from a family of expanders of super-constant degree and super-constant girth.
The method of Batson, Spielman and Srivastava proves a more general result,
about sparsifying sums of rank-one matrices, and their method applies to an
"online" setting. We show that for the online matrix setting the bound is tight, up to lower order terms
An Alon-Boppana type bound for weighted graphs and lowerbounds for spectral sparsification
No abstract availabl
Cut Sparsification of the Clique Beyond the Ramanujan Bound: A Separation of Cut Versus Spectral Sparsification
We prove that a random -regular graph, with high probability, is a cut
sparsifier of the clique with approximation error at most , where
and denotes an error term that depends on and and goes to
zero if we first take the limit and then the limit .
This is established by analyzing linear-size cuts using techniques of
Jagannath and Sen derived from ideas in statistical physics, and analyzing
small cuts via martingale inequalities.
We also prove new lower bounds on spectral sparsification of the clique. If
is a spectral sparsifier of the clique and has average degree , we
prove that the approximation error is at least the "Ramanujan bound''
, which is met by -regular Ramanujan graphs,
provided that either the weighted adjacency matrix of is a (multiple of) a
doubly stochastic matrix, or that satisfies a certain high "odd
pseudo-girth" property. The first case can be seen as an "Alon-Boppana theorem
for symmetric doubly stochastic matrices," showing that a symmetric doubly
stochastic matrix with non-zero entries has a non-trivial eigenvalue of
magnitude at least ; the second case generalizes a
lower bound of Srivastava and Trevisan, which requires a large girth
assumption.
Together, these results imply a separation between spectral sparsification
and cut sparsification. If is a random -regular graph on
vertices, we show that, with high probability, admits a (weighted subgraph)
cut sparsifier of average degree and approximation error at most
, while every (weighted subgraph) spectral
sparsifier of having average degree has approximation error at least
.Comment: To appear in SODA 202
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum