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 rigorous...

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Bibliographic Details
Published in:Discrete mathematics and theoretical computer science Vol. DMTCS Proceedings vol. AH,...; no. Proceedings; pp. 95 - 108
Main Authors: Coja-Oghlan, Amin, Krivelevich, Michael, Vilenchik, Dan
Format: Journal Article Conference Proceeding
Language:English
Published: DMTCS 01.01.2007
Discrete Mathematics and Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science
Series:DMTCS Proceedings
Subjects:
[info.info-cg] computer science [cs]/computational geometry [cs.cg]
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds]
[math.math-co] mathematics [math]/combinatorics [math.co]
algorithms and data structures
Combinatorics
computational and structural complexity
Computational Geometry
Computer Science
Data Structures and Algorithms
Discrete Mathematics
Mathematics
message passing algorithms
sat
message passing algorithms
algorithms and data structures
SAT
computational and structural complexity
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary: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.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.3538