15 research outputs found
Random k-SAT and the Power of Two Choices
We study an Achlioptas-process version of the random k-SAT process: a bounded
number of k-clauses are drawn uniformly at random at each step, and exactly one
added to the growing formula according to a particular rule. We prove the
existence of a rule that shifts the satisfiability threshold. This extends a
well-studied area of probabilistic combinatorics (Achlioptas processes) to
random CSP's. In particular, while a rule to delay the 2-SAT threshold was
known previously, this is the first proof of a rule to shift the threshold of
k-SAT for k >= 3.
We then propose a gap decision problem based upon this semi-random model. The
aim of the problem is to investigate the hardness of the random k-SAT decision
problem, as opposed to the problem of finding an assignment or certificate of
unsatisfiability. Finally, we discuss connections to the study of Achlioptas
random graph processes.Comment: 13 page
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution to select
variable , where is a parameter of the model. This results into
formulas where the number of occurrences of variables follows a power-law
distribution where . This property
has been observed in most real-world SAT instances. For , our model
extends classical random SAT instances.
We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with when the clause/variable
ratio is . We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds . %This implies that the SAT/UNSAT
phase transition phenomena vanishes when , and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when , the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties similar to
real-world instances. It is known that many industrial instances, even with a great number of
variables, can be solved by a clever solver in a reasonable amount of time. This is not possible,
in general, with classical randomly generated instances. We provide a different generation model
of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform
probability distribution P(i) ∼ i
−β
to select variable i, where β is a parameter of the model. This
results in formulas where the number of occurrences k of variables follows a power-law distribution
P(k) ∼ k
−δ
, where δ = 1 + 1/β. This property has been observed in most real-world SAT instances.
For β = 0, our model extends classical random SAT instances. We prove the existence of a SAT–
UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β < 1/2 when
the clause/variable ratio is m/n =
1−2β
(1−β)
2
. We also prove that scale-free random k-SAT instances are
unsatisfiable with a high probability when the number of clauses exceeds ω(n
(1−β)k
). The proof of
this result suggests that, when β > 1 − 1/k, the unsatisfiability of most formulas may be due to small
cores of clauses. Finally, we show how this model will allow us to generate random instances similar
to industrial instances, of interest for testing purposes.This research was supported by the project PROOFS, Grant PID2019-109137GB-C21 funded by MCIN/AEI/10.13039/501100011033
Given enough choice, simple local rules percolate discontinuously
There is still much to discover about the mechanisms and nature of
discontinuous percolation transitions. Much of the past work considers graph
evolution algorithms known as Achlioptas processes in which a single edge is
added to the graph from a set of randomly chosen candidate edges at each
timestep until a giant component emerges. Several Achlioptas processes seem to
yield a discontinuous percolation transition, but it was proven by Riordan and
Warnke that the transition must be continuous in the thermodynamic limit.
However, they also proved that if the number of candidate edges
increases with the number of nodes, then the percolation transition may be
discontinuous. Here we attempt to find the simplest such process which yields a
discontinuous transition in the thermodynamic limit. We introduce a process
which considers only the degree of candidate edges and not component size. We
calculate the critical point and rigorously
show that the critical window is of size . If grows
very slowly, for example , the critical window is barely sublinear
and hence the phase transition is discontinuous but appears continuous in
finite systems. We also present arguments that Achlioptas processes with
bounded size rules will always have continuous percolation transitions even
with infinite choice.Comment: Accepted to European Physical Journal
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