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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Vydáno v:	Discrete mathematics and theoretical computer science Ročník DMTCS Proceedings vol. AH,...; číslo Proceedings; s. 95 - 108
Hlavní autoři:	Coja-Oghlan, Amin, Krivelevich, Michael, Vilenchik, Dan
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	DMTCS 01.01.2007 Discrete Mathematics and Theoretical Computer Science Discrete Mathematics & Theoretical Computer Science
Edice:	DMTCS Proceedings
Témata:	[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
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Popis
Shrnutí:	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