Why almost all satisfiable $k$-CNF formulas are easy - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2007

Why almost all satisfiable $k$-CNF formulas are easy


Finding a satisfying assignment for a $k$-CNF formula $(k \geq 3)$, assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable $k$-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though linear, number of variables. This observation enables us to describe a polynomial time algorithm that finds $\textit{whp}$ a satisfying assignment for such formulas, thus asserting that most satisfiable $k$-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse $k$-CNF formulas (which are satisfiable $\textit{whp}$), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more "regular" structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation.
Fichier principal
Vignette du fichier
dmAH0107.pdf (257.28 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive

Dates and versions

hal-01184786 , version 1 (17-08-2015)



Amin Coja-Oghlan, Michael Krivelevich, Dan Vilenchik. Why almost all satisfiable $k$-CNF formulas are easy. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. pp.95-108, ⟨10.46298/dmtcs.3538⟩. ⟨hal-01184786⟩


295 View
1064 Download



Gmail Facebook Twitter LinkedIn More