Karloff–Zwick algorithm: Difference between revisions

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The '''Karloff–Zwick algorithm''', in [[computational complexity theory]], is a [[randomized algorithm|randomised]] [[approximation algorithm]] taking an instance of [[MAX-3SAT]] [[Boolean satisfiability problem]] as input. If the instance is satisfiable, then the expected weight of the assignment found is at least 7/8 of optimal. ItThere providesis strong evidence (but not a [[mathematical proof]]) that the algorithm performsachieves equally7/8 of optimal welleven on arbitraryunsatisfiable MAX-3SAT instances. [[Howard Karloff]] and [[Uri Zwick]] presented the algorithm in 1997.<ref name="Karloff">{{citation|last1=Karloff|first1= H.|title= Proceedings 38th Annual Symposium on Foundations of Computer Science|last2= Zwick|first2= U. |contributionchapter=A 7/8-approximation algorithm for MAX 3SAT?|title-link=[[Symposium on Foundations of Computer Science]]|title= Proc. 38th Annual Symposium on Foundations of Computer Science|year=1997|pages=406–415|doi=10.1109/SFCS.1997.646129|isbn= 978-0-8186-8197-4|citeseerx= 10.1.1.51.1351|s2cid= 15447333}}.</ref>

The algorithm is based on [[semidefinite programming]]. It can be derandomized using, e.g., the techniques from <ref>{{citation|last1=Sivakumar |first1=D. |title=Proceedings of the thiry-fourth annual ACM symposium on Theory of computing |chapter=Algorithmic derandomization via complexity theory |date=19 May 2002 |pages=619–626 |doi=10.1145/509907.509996|isbn=1581134959 |s2cid=94045 }}</ref> to yield a deterministic [[polynomial-time]] algorithm with the same approximation guarantees.

==Comparison to random assignment==

For the related MAX-E3SAT problem, in which all clauses in the input 3SAT formula are guaranteed to have exactly three literals, the simple [[randomized algorithm|randomized]] [[approximation algorithm]] which assigns a truth value to each variable independently and uniformly at random satisfies 7/8 of all clauses in expectation, irrespective of whether the original formula is satisfiable. Further, this simple algorithm can also be easily [[Randomized_algorithm#Derandomization|derandomized]] using the [[Method_of_conditional_probabilities#The_method_of_conditional_probabilities_with_conditional_expectations|method of conditional expectations]]. The Karloff–Zwick algorithm, however, does not require the restriction that the input formula should have three literals in every clause.<ref name="Karloff"/>

==Optimality==

Building upon previous work on the [[PCP theorem]], [[Johan Håstad]] showed that, assuming P ≠ NP, no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7/8, even when restricted to satisfiable instances of the problem in which each clause contains exactly three literals. Both the Karloff–Zwick algorithm and the above simple algorithm are therefore optimal in this sense.<ref>{{citation|first=J.|last=Hastad|title=Some optimal inapproximability results|journal=[[Journal of the ACM]]|volume=48|issue=4|year=2001|doi=10.1145/502090.502098|pages=798–859|citeseerx=10.1.1.638.2808|s2cid=5120748 }}.</ref>

== References ==

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[[Category:Approximation algorithms]]

[[Category:Randomized algorithms]]

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