Is Max 3-SAT a W[1] hard problem, parmeterized by some parmeterize? I can't find the relevant literature.

I accept any parameterization.

  • $\begingroup$ Is the question whether there exists any parameter at all that makes MaxSat W[1]-hard? $\endgroup$
    – Pål GD
    Nov 5 '21 at 17:36
  • $\begingroup$ Like the parameter 1? $\endgroup$
    – Pål GD
    Nov 5 '21 at 17:50
  • $\begingroup$ Please clarify what you mean. Do you accept any parameterization? Or do you mean that max-parameter? $\endgroup$
    – Pål GD
    Dec 7 '21 at 22:50

The W hierarchy consists of classes of decision problems. The decision versions of MAX-SAT problems are just slightly disguised SAT problems. For example, to transform the decision problem "is there a variable assignment that satisfies k clauses?", add a unique variable to each CNF clause, then add a constraint that demands that at least k of the unique variables be set false. (All this can be done in polynomial time.) If the resulting SAT instance is satisfiable, the answer to the MAX-SAT problem is yes, otherwise the answer is no.

Since SAT Karp-reduces to CLIQUE and CLIQUE is W[1]-hard, SAT must be as well, and therefore so is the decision version of MAX-SAT.

  • $\begingroup$ SAT Karp-reduces to CLIQUE cannot show that SAT must be W[1] hard. If clique can reduce to SAT, it is right. Is my understanding correct? $\endgroup$
    – zhukui bai
    Nov 7 '21 at 3:28
  • $\begingroup$ So MAX SAT is W-hard for what parameter? Also, as an example, vertex cover reduces to clique, yet VC is not W-hard (for the natural parameter). $\endgroup$
    – Juho
    Nov 7 '21 at 8:14

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