Parameterized Complexity of MaxSat above Average

In MaxSat, we are given a CNF formula F with n variables and m clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let r1, . . . , rm be the number of literals in the clauses of F . Then asat(F ) = ∑m i=1(1 − 2 −ri) is the expected number of clauses satisfied by a random truth assignment (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 MaxLin2-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 rj ≤ 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 r(n) = dlogne. We also prove that assuming the exponential time hypothesis, Max-r(n)-Sat-AA is not in XP already for any r(n) ≥ log logn + φ(n), where φ(n) is any unbounded strictly increasing 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) ≤ log log n − log log logn and (ii) fixed-parameter tractable for any r(n) ≤ log logn − log log logn − φ(n), where φ(n) is any unbounded strictly increasing function. The proof uses some results on MaxLin2-AA.