On MAX–SAT with cardinality constraint

We consider the weighted MAX–SAT problem with an additional constraint that at most k variables can be set to true. We call this problem k–WMAX–SAT. This problem admits a $(1-\frac{1}{e})$-factor approximation algorithm in polynomial time [Sviridenko, Algorithmica (2001)] and it is proved that there is no $(1-\frac{1}{e}+ϵ)$-factor approximation algorithm in $f\left(k\right)\cdot n^{o\left(k\right)}$ time for Maximum Coverage, the unweighted monotone version of k–WMAX–SAT [Manurangsi, SODA 2020]. Therefore, we study two restricted versions of the problem in the realm of parameterized complexity.

• 1.
  When the input is an unweighted 2–CNF formula (the problem is called k–MAX–2SAT), we design an efficient polynomial-size approximate kernelization scheme. That is, we design a polynomial-time algorithm that given a 2–CNF formula ψ and $ϵ>0$, compresses the input instance to a 2–CNF formula ${\psi }^{\star }$ such that any c-approximate solution of ${\psi }^{\star }$ can be converted to a $c(1-ϵ)$-approximate solution of ψ in polynomial time.
• 2.
  When the input is a planar CNF formula, i.e., the variable-clause incidence graph is a planar graph, we show the following results:

  • •
    There is an FPT algorithm for k–WMAX–SAT on planar CNF formulas that runs in $2^{O\left(k\right)}\cdot (C+V)$ time.
  • •
    There is a polynomial-time approximation scheme for k–WMAX–SAT on planar CNF formulas that runs in time $2^{O\left(\frac{1}{ϵ}\right)}\cdot k\cdot (C+V)$.

  The above-mentioned C and V are the number of clauses and variables of the input formula respectively.