23 research outputs found
Sum-of-squares lower bounds for planted clique
Finding cliques in random graphs and the closely related "planted" clique
variant, where a clique of size k is planted in a random G(n, 1/2) graph, have
been the focus of substantial study in algorithm design. Despite much effort,
the best known polynomial-time algorithms only solve the problem for k ~
sqrt(n).
In this paper we study the complexity of the planted clique problem under
algorithms from the Sum-of-squares hierarchy. We prove the first average case
lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the
SOS hierarchy cannot find a planted k-clique unless k > n^{1/2r} (up to
logarithmic factors). Thus, for any constant number of rounds planted cliques
of size n^{o(1)} cannot be found by this powerful class of algorithms. This is
shown via an integrability gap for the natural formulation of maximum clique
problem on random graphs for SOS and Lasserre hierarchies, which in turn follow
from degree lower bounds for the Positivestellensatz proof system.
We follow the usual recipe for such proofs. First, we introduce a natural
"dual certificate" (also known as a "vector-solution" or "pseudo-expectation")
for the given system of polynomial equations representing the problem for every
fixed input graph. Then we show that the matrix associated with this dual
certificate is PSD (positive semi-definite) with high probability over the
choice of the input graph.This requires the use of certain tools. One is the
theory of association schemes, and in particular the eigenspaces and
eigenvalues of the Johnson scheme. Another is a combinatorial method we develop
to compute (via traces) norm bounds for certain random matrices whose entries
are highly dependent; we hope this method will be useful elsewhere
Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems
Given a large data matrix , we consider the
problem of determining whether its entries are i.i.d. with some known marginal
distribution , or instead contains a principal submatrix
whose entries have marginal distribution . As a special case, the hidden (or planted) clique problem
requires to find a planted clique in an otherwise uniformly random graph.
Assuming unbounded computational resources, this hypothesis testing problem
is statistically solvable provided for a suitable
constant . However, despite substantial effort, no polynomial time algorithm
is known that succeeds with high probability when .
Recently Meka and Wigderson \cite{meka2013association}, proposed a method to
establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy.
Here we consider the degree- SOS relaxation, and study the construction of
\cite{meka2013association} to prove that SOS fails unless . An argument presented by Barak implies that this lower bound
cannot be substantially improved unless the witness construction is changed in
the proof. Our proof uses the moments method to bound the spectrum of a certain
random association scheme, i.e. a symmetric random matrix whose rows and
columns are indexed by the edges of an Erd\"os-Renyi random graph.Comment: 40 pages, 1 table, conferenc
Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs
We study subgraphs of Paley graphs of prime order induced on the sets of
vertices extending a given independent set of size to a larger independent
set. Using a sufficient condition proved in the author's recent companion work,
we show that a family of character sum estimates would imply that, as , the empirical spectral distributions of the adjacency matrices of any
sequence of such subgraphs have the same weak limit (after rescaling) as those
of subgraphs induced on a random set including each vertex independently with
probability , namely, a Kesten-McKay law with parameter . We prove
the necessary estimates for , obtaining in the process an alternate
proof of a character sum equidistribution result of Xi (2022), and provide
numerical evidence for this weak convergence for . We also conjecture
that the minimum eigenvalue of any such sequence converges (after rescaling) to
the left edge of the corresponding Kesten-McKay law, and provide numerical
evidence for this convergence. Finally, we show that, once , this
(conjectural) convergence of the minimum eigenvalue would imply bounds on the
clique number of the Paley graph improving on the current state of the art due
to Hanson and Petridis (2021), and that this convergence for all
would imply that the clique number is .Comment: 43 pages, 1 table, 6 figure
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum