Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

Abstract

For a finite set Γ of Boolean relations, M ax O nes SAT(Γ) and E xact O nes SAT(Γ) are generalized satisfiability problems where every constraint relation is from Γ, and the task is to find a satisfying assignment with at least/exactly k variables set to 1, respectively. We study the parameterized complexity of these problems, including the question whether they admit polynomial kernels. For M ax O nes SAT(Γ), we give a classification into five different complexity levels: polynomial-time solvable, admits a polynomial kernel, fixed-parameter tractable, solvable in polynomial time for fixed k , and NP-hard already for k = 1. For E xact O nes SAT(Γ), we refine the classification obtained earlier by taking a closer look at the fixed-parameter tractable cases and classifying the sets Γ for which E xact O nes SAT(Γ) admits a polynomial kernel.