383 research outputs found
Pairs of SAT Assignment in Random Boolean Formulae
We investigate geometrical properties of the random K-satisfiability problem
using the notion of x-satisfiability: a formula is x-satisfiable if there exist
two SAT assignments differing in Nx variables. We show the existence of a sharp
threshold for this property as a function of the clause density. For large
enough K, we prove that there exists a region of clause density, below the
satisfiability threshold, where the landscape of Hamming distances between SAT
assignments experiences a gap: pairs of SAT-assignments exist at small x, and
around x=1/2, but they donot exist at intermediate values of x. This result is
consistent with the clustering scenario which is at the heart of the recent
heuristic analysis of satisfiability using statistical physics analysis (the
cavity method), and its algorithmic counterpart (the survey propagation
algorithm). The method uses elementary probabilistic arguments (first and
second moment methods), and might be useful in other problems of computational
and physical interest where similar phenomena appear
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
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
On the size of irredundant propagation complete CNF formulas
We investigate propagation complete (PC) CNF formulas for a symmetric
definite Horn function of variables and demonstrate that the minimum size
of these formulas is closely related to specific covering numbers, namely, to
the smallest number of -subsets of an -set covering all -subsets
for a suitable . As a consequence, we demonstrate an irredundant PC formula
whose size is larger than the size of a smallest PC formula for the same
function by a factor . This complements a known polynomial
upper bound on this factor.Comment: 16 page
Randomness Extraction in AC0 and with Small Locality
Randomness extractors, which extract high quality (almost-uniform) random
bits from biased random sources, are important objects both in theory and in
practice. While there have been significant progress in obtaining near optimal
constructions of randomness extractors in various settings, the computational
complexity of randomness extractors is still much less studied. In particular,
it is not clear whether randomness extractors with good parameters can be
computed in several interesting complexity classes that are much weaker than P.
In this paper we study randomness extractors in the following two models of
computation: (1) constant-depth circuits (AC0), and (2) the local computation
model. Previous work in these models, such as [Vio05a], [GVW15] and [BG13],
only achieve constructions with weak parameters. In this work we give explicit
constructions of randomness extractors with much better parameters. As an
application, we use our AC0 extractors to study pseudorandom generators in AC0,
and show that we can construct both cryptographic pseudorandom generators
(under reasonable computational assumptions) and unconditional pseudorandom
generators for space bounded computation with very good parameters.
Our constructions combine several previous techniques in randomness
extractors, as well as introduce new techniques to reduce or preserve the
complexity of extractors, which may be of independent interest. These include
(1) a general way to reduce the error of strong seeded extractors while
preserving the AC0 property and small locality, and (2) a seeded randomness
condenser with small locality.Comment: 62 page
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