Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

Abstract

AbstractFor a finite set \(\it\Gamma\) of Boolean relations, Max Ones SAT(\(\it\Gamma\)) and Exact Ones SAT(\(\it\Gamma\)) are generalized satisfiability problems where every constraint relation is from \(\it\Gamma\), 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 Max Ones SAT(\(\it\Gamma\)), we give a classification into 5 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 Exact Ones SAT(\(\it\Gamma\)), we refine the classification obtained earlier by having a closer look at the fixed-parameter tractable cases and classifying the sets \(\it\Gamma\) for which Exact Ones SAT(\(\it\Gamma\)) admits a polynomial kernel.KeywordsPolynomial TimeParameterized ComplexityConstraint Satisfaction ProblemVertex CoverPolynomial KernelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.