NDLI: On the Diameter of the Set of Satisfying Assignments in Random Satisfiable k-CNF Formulas

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Feige, Uriel Flaxman, Abraham D. Vilenchik, Dan
Copyright Year	2011
Abstract	It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2kln2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable, whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas and inspect another parameterthe diameter of the solution space (that is, the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n/2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisely, at ratio (1+ε)2kln2, ε=ε(k) tending to 0 as k grows) the diameter is almost surely 𝒪(k2-kn). This shows that a relatively small change in the density around the satisfiability threshold (a multiplicative (1+ε) factor) makes a dramatic change in the diameter. This drop in the diameter cannot be attributed to the fact that a larger fraction of the formulas are not satisfiable (and hence have diameter 0), because the nonsatisfiable formulas are excluded from consideration by our conditioning that the formula be satisfiable.
Starting Page	736
Ending Page	749
Page Count	14
File Format	PDF
ISSN	08954801
DOI	10.1137/090749323
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-07-01
Access Restriction	Subscribed
Subject Keyword	phase transition random k-SAT
Content Type	Text
Resource Type	Article
Subject	Mathematics

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