59 research outputs found
Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
We present an algorithm for recovering planted solutions in two well-known
models, the stochastic block model and planted constraint satisfaction
problems, via a common generalization in terms of random bipartite graphs. Our
algorithm matches up to a constant factor the best-known bounds for the number
of edges (or constraints) needed for perfect recovery and its running time is
linear in the number of edges used. The time complexity is significantly better
than both spectral and SDP-based approaches.
The main contribution of the algorithm is in the case of unequal sizes in the
bipartition (corresponding to odd uniformity in the CSP). Here our algorithm
succeeds at a significantly lower density than the spectral approaches,
surpassing a barrier based on the spectral norm of a random matrix.
Other significant features of the algorithm and analysis include (i) the
critical use of power iteration with subsampling, which might be of independent
interest; its analysis requires keeping track of multiple norms of an evolving
solution (ii) it can be implemented statistically, i.e., with very limited
access to the input distribution (iii) the algorithm is extremely simple to
implement and runs in linear time, and thus is practical even for very large
instances
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
Satisfiability Modulo Transcendental Functions via Incremental Linearization
In this paper we present an abstraction-refinement approach to Satisfiability
Modulo the theory of transcendental functions, such as exponentiation and
trigonometric functions. The transcendental functions are represented as
uninterpreted in the abstract space, which is described in terms of the
combined theory of linear arithmetic on the rationals with uninterpreted
functions, and are incrementally axiomatized by means of upper- and
lower-bounding piecewise-linear functions. Suitable numerical techniques are
used to ensure that the abstractions of the transcendental functions are sound
even in presence of irrationals. Our experimental evaluation on benchmarks from
verification and mathematics demonstrates the potential of our approach,
showing that it compares favorably with delta-satisfiability /interval
propagation and methods based on theorem proving
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
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
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