Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

Abstract

For 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.

The second author is supported by ERC Advanced Grant DMMCA and the Hungarian National Research Fund (OTKA grant 67651).