Testing satisfiability

Noga Alon, Asaf Shapira*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let Φ be a set of general boolean functions on n variables, such that each function depends on exactly k variables, and each variable can take a value from [1, d]. We say that Φ is ε-far from satisfiable, if one must remove at least εnk functions in order to make the set of remaining functions satisfiable. Our main result is that if Φ is ε-far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k, d)/ε2, are not satisfiable, where c(k, d) depends only on k and d. Using the above claim, we obtain similar results for k-SAT and k-NAEQ-SAT. Assume we relax the decision problem of whether an instance of one of the above mentioned problems is satisfiable or not, to the problem of deciding whether an instance is satisfiable or ε-far from satisfiable. While the above decision problems are NP-hard, our result implies that we can solve their relaxed versions, that is, distinguishing between satisfiable and ε-far from satisfiable instances, in randomized constant time. From the above result we obtain as a special case, previous results of Alon and Krivelevich, and of Czumaj and Sohler, concerning testing of graphs and hypergraphs colorability. We also discuss the difference between testing with one-sided and two-sided error.

Original languageEnglish
Pages (from-to)87-103
Number of pages17
JournalJournal of Algorithms
Issue number2
StatePublished - Jul 2003


FundersFunder number
Deutsch institute
Hermann Minkowski Minerva Center for Geometry
Bloom's Syndrome Foundation
Israel Science Foundation
Tel Aviv University


    • Hypergraph coloring
    • Property testing
    • Satisfiability


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