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{{expert-subject|Computer science|ex2=Mathematics|date=May 2009}}
{{Lead too short|date=May 2009}}
'''MAXEkSAT''' is a problem in [[computational complexity theory]] that is a maximization version of the Boolean satisfiability problem [[3SAT]].


In MAXEkSAT, each clause has exactly ''k'' literals, with ''k'' ≥ 3, and is in [[conjunctive normal form]].  These formulas are called kCNF formulas.


The problem is to determine the maximum number of clauses that can be satisfied by a truth assignment to the variables in the clauses.
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We say that an algorithm ''A'' provides an ''α''-[[approximation algorithm|approximation]] to MAXEkSAT if, for some fixed positive ''α'' less than or equal to 1, and every kCNF formula ''φ'', ''A'' can find a truth assignment to the variables of ''φ'' that will satisfy at least an ''α''-fraction of the maximum number of satisfiable clauses of ''φ''.
 
Note that since 3SAT is [[NP-hard]], and since any 1-approximation algorithm for MAXEkSAT could easily be converted into an algorithm for accepting the language 3SAT, 1-approximation of MAXEkSAT is NP-hard.
 
A natural next question is how big ''α'' can be guaranteed to be for an explicit algorithm that runs in polynomial time ('''[[P (complexity)|P]]''').
 
==Main idea==
A simple result is that there is a simple randomized algorithm that provides a 7/8-approximation to MAXE3SAT which runs in polynomial time: For each ''i'', set variable ''x''<sub>''i''</sub> in formula ''φ'' to be true with probability 1/2, and false with probability 1/2.
 
==Proof==
In a 3CNF formula, there are three boolean variables; consequently, the number of possible combination of assignments to the three variables is 8. Since there is only one combination of truth assignments to a clause that will render it false, we have a 7/8 probability of stumbling across a valid assignment for any given clause.  Thus, the expected fraction of satisfied clauses is 7/8. Because this expectation is over all possible assignments to variables, this guarantees the existence of some assignment that satisfies at least 7/8-fraction of clauses of any 3CNF formula.
 
==Main question==
But even if we know that there exists such a satisfying assignment, how would we find one?  We could cycle through all 2<sup>''n''</sup> assignments until we found one, but clearly that approach can't be guaranteed to run in polynomial time.
 
One answer, relying on results in the study of [[error correcting codes]], works for any constant ''k'', providing a polynomial time algorithm for finding such an assignment.
 
We need one definition and two facts to find the algorithm.
 
==Definition==
<math>S\subseteq\{0,1\}^n</math> is an ''ℓ''-wise independent source if, for a uniformly chosen random (''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>)&nbsp;∈&nbsp;''S'', ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> are [[pairwise independence|''ℓ''-wise independent]] random variables.
 
==Fact 1==
Note that such an assignment can be found among elements of any ''ℓ''-wise independent source over ''n'' binary variables. This is easier to see once you realize that an ''ℓ''-wise independent source is really just any set of binary vectors over {0,&nbsp;1}<sup>''n''</sup> with the property that all restrictions of those vectors to ''ℓ'' co-ordinates must present the 2<sup>''ℓ''</sup> possible binary combinations an equal number of times.
 
==Fact 2==
Recall that BCH<sub>2,''m'',''d''</sub> is an <math> [n=2^m, n-1 -\lceil {d-2}/2\rceil m, d]_2</math> linear code.
 
There exists an ''ℓ''-wise independent source of size <math>O(n^{\lfloor \ell/2 \rfloor})</math>, namely the dual of a BCH<sub>2,log&nbsp;''n'',''ℓ''+1</sub> code, which is a linear code.  Since every [[BCH code]] can be presented as  a polynomial-time computable restriction of a related [[Reed Solomon]] code, which itself is strongly explicit, there is a polynomial-time algorithm for finding such an assignment to the ''x''<sub>''i''</sub>'s. The proof of fact 2 can be found at [[Dual of BCH is an independent source]].
 
To summarize, generate BCH<sub>2,log&nbsp;''n'',''ℓ''+1</sub>, compute its dual, which as a set is an ''ℓ''-wise independent source, and treat each element (codeword) of that source as a truth assignment to the ''n'' variables in ''φ''.  At least one of them will satisfy at least 1&nbsp;&minus;&nbsp;2<sup>&minus;''ℓ''</sup> of the clauses of ''φ'', whenever ''φ'' is in kCNF form, ''k''&nbsp;=&nbsp;''ℓ''.
 
For ''ℓ''&nbsp;=&nbsp;3, this derandomizes the initially described algorithm for MAXE3SAT.
 
== Related problems ==
[[MAX3SAT]] is a relaxed version of MAXEkSAT, where each clause can have no more than three literals.
 
== References ==
{{reflist}}
*[http://www.cse.buffalo.edu/~atri/courses/coding-theory/ Coding Theory notes at University at Buffalo]
*[http://people.csail.mit.edu/madhu/FT01/course.html Coding Theory notes at MIT]
 
[[Category:NP-hard problems]]

Revision as of 06:49, 5 March 2014


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