49 research outputs found
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
Sum of squares lower bounds for refuting any CSP
Let be a nontrivial -ary predicate. Consider a
random instance of the constraint satisfaction problem on
variables with constraints, each being applied to randomly
chosen literals. Provided the constraint density satisfies , such
an instance is unsatisfiable with high probability. The \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate supports a -\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree
(which runs in time ) \emph{cannot} refute a random instance of
. In particular, the polynomial-time SOS algorithm requires
constraints to refute random instances of
CSP when supports a -wise uniform distribution on its satisfying
assignments. Together with recent work of Lee et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least constraints.
Our results (which also extend with no change to CSPs over larger alphabets)
subsume all previously known lower bounds for semialgebraic refutation of
random CSPs. For every constraint predicate~, they give a three-way hardness
tradeoff between the density of constraints, the SOS degree (hence running
time), and the strength of the refutation. By recent algorithmic results of
Allen et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur
Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"
We present an algorithm for strongly refuting smoothed instances of all
Boolean CSPs. The smoothed model is a hybrid between worst and average-case
input models, where the input is an arbitrary instance of the CSP with only the
negation patterns of the literals re-randomized with some small probability.
For an -variable smoothed instance of a -arity CSP, our algorithm runs in
time, and succeeds with high probability in bounding the optimum
fraction of satisfiable constraints away from , provided that the number of
constraints is at least . This
matches, up to polylogarithmic factors in , the trade-off between running
time and the number of constraints of the state-of-the-art algorithms for
refuting fully random instances of CSPs [RRS17].
We also make a surprising new connection between our algorithm and even
covers in hypergraphs, which we use to positively resolve Feige's 2008
conjecture, an extremal combinatorics conjecture on the existence of even
covers in sufficiently dense hypergraphs that generalizes the well-known Moore
bound for the girth of graphs. As a corollary, we show that polynomial-size
refutation witnesses exist for arbitrary smoothed CSP instances with number of
constraints a polynomial factor below the "spectral threshold" of ,
extending the celebrated result for random 3-SAT of Feige, Kim and Ofek
[FKO06]
Certifying solution geometry in random CSPs: counts, clusters and balance
An active topic in the study of random constraint satisfaction problems
(CSPs) is the geometry of the space of satisfying or almost satisfying
assignments as the function of the density, for which a precise landscape of
predictions has been made via statistical physics-based heuristics. In
parallel, there has been a recent flurry of work on refuting random constraint
satisfaction problems, via nailing refutation thresholds for spectral and
semidefinite programming-based algorithms, and also on counting solutions to
CSPs. Inspired by this, the starting point for our work is the following
question: what does the solution space for a random CSP look like to an
efficient algorithm?
In pursuit of this inquiry, we focus on the following problems about random
Boolean CSPs at the densities where they are unsatisfiable but no refutation
algorithm is known.
1. Counts. For every Boolean CSP we give algorithms that with high
probability certify a subexponential upper bound on the number of solutions. We
also give algorithms to certify a bound on the number of large cuts in a
Gaussian-weighted graph, and the number of large independent sets in a random
-regular graph.
2. Clusters. For Boolean CSPs we give algorithms that with high
probability certify an upper bound on the number of clusters of solutions.
3. Balance. We also give algorithms that with high probability certify that
there are no "unbalanced" solutions, i.e., solutions where the fraction of
s deviates significantly from .
Finally, we also provide hardness evidence suggesting that our algorithms for
counting are optimal
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
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
CSP-Completeness And Its Applications
We build off of previous ideas used to study both reductions between CSPrefutation problems and improper learning and between CSP-refutation problems themselves to expand some hardness results that depend on the assumption that refuting random CSP instances are hard for certain choices of predicates (like k-SAT). First, we are able argue the hardness of the fundamental problem of learning conjunctions in a one-sided PAC-esque learning model that has appeared in several forms over the years. In this model we focus on producing a hypothesis that foremost guarantees a small false-positive rate while minimizing the false-negative rate for such hypotheses. Further, we formalize a notion of CSP-refutation reductions and CSP-refutation completeness that and use these, along with candidate CSP-refutatation complete predicates, to provide further evidence for the hardness of several problems