Parameterized complexity of MaxSat Above Average

  • Authors:

  • Robert Crowston;Gregory Gutin;Mark Jones;Venkatesh Raman;Saket Saurabh

  • Affiliations:

  • Royal Holloway, University of London, Egham, Surrey, UK;Royal Holloway, University of London, Egham, Surrey, UK;Royal Holloway, University of London, Egham, Surrey, UK;The Institute of Mathematical Sciences, Chennai 600 113, India;The Institute of Mathematical Sciences, Chennai 600 113, India

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

In MaxSat, we are given a CNF formula F with n variables and m clauses; the task is to find a truth assignment satisfying the maximum number of clauses. Let r"1,...,r"m be the number of literals in the clauses of F. Then asat(F)=@?"i"="1^m(1-2^-^r^"^i) is the expected number of clauses satisfied by a random truth assignment, when the truth values to the variables are distributed uniformly and independently. It is well-known that, in polynomial time, one can find a truth assignment satisfying at least asat(F) clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least asat(F)+k clauses, where k is the (nonnegative) parameter. We prove that MaxSat-AA is para-NP-complete and thus, MaxSat-AA is not fixed-parameter tractable unless P=NP. This is in sharp contrast to the similar problem MaxLin 2-AA which was recently proved to be fixed-parameter tractable by Crowston et al. (FSTTCS 2011). In fact, we consider a more refined version of MaxSat-AA, Max-r(n)-Sat-AA, where r"j@?r(n) for each j. Alon et al. (SODA 2010) proved that if r=r(n) is a constant, then Max-r-Sat-AA is fixed-parameter tractable. We prove that Max-r(n)-Sat-AA is para-NP-complete for any r(n)=@?logn@?. We also prove that assuming the exponential time hypothesis, Max-r(n)-Sat-AA is not even in XP for any integral r(n)=loglogn+@f(n), where @f(n) is any real-valued unbounded strictly increasing computable function. This lower bound on r(n) cannot be decreased much further as we prove that Max-r(n)-Sat-AA is (i) in XP for any r(n)@?loglogn-logloglogn and (ii) fixed-parameter tractable for any r(n)@?loglogn-logloglogn-@f(n), where @f(n) is any real-valued unbounded strictly increasing computable function. The proof uses some results on MaxLin 2-AA.