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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
On the random satisfiable process
In this work we suggest a new model for generating random satisfiable k-CNF
formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k}
possible clauses over the variables x_1, ..., x_n, and starting from the empty
formula, go over the clauses one by one, including each new clause as you go
along if after its addition the formula remains satisfiable. We study the
evolution of this process, namely the distribution over formulas obtained after
scanning through the first m clauses (in the random permutation's order).
Random processes with conditioning on a certain property being respected are
widely studied in the context of graph properties. This study was pioneered by
Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and
also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite
graphs. Since then many other graph properties were studied such as planarity
and H-freeness. Thus our model is a natural extension of this approach to the
satisfiability setting.
Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently
large constant, we are able to characterize the structure of the solution space
of a typical formula in this distribution. Specifically, we show that typically
all satisfying assignments are essentially clustered in one cluster, and all
but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying
assignments. We also describe a polynomial time algorithm that finds with high
probability a satisfying assignment for such formulas
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