Parameterized complexity of MaxSat Above Average

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 r1,…,rm be the number of literals in the clauses of F. Then 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 clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least 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 . 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 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 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+ϕ(n), where ϕ(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−ϕ(n), where ϕ(n) is any real-valued unbounded strictly increasing computable function. The proof uses some results on MaxLin 2-AA.