Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints
<|reference_start|>Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints: In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $\Phi$, and $k \ge 0$, and the goal is to find an assignment $\beta$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $\beta$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $\Phi$ is monotone, i.e., does not contain any negative literals. CC-MaxSAT and MaxCov are extremely well-studied problems in the approximation algorithms as well as parameterized complexity literature.   Our first contribution is that the two problems are equivalent to each other in the context of FPT-Approximation parameterized by $k$ (approximation is in terms of number of clauses satisfied/elements covered). We give a randomized reduction from CC-MaxSAT to MaxCov in time $O(1/\epsilon)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-\epsilon$. Furthermore, this reduction also works in the presence of fairness and matroid constraints.   Armed with this reduction, we focus on designing FPT-Approximation schemes (FPT-ASes) for MaxCov and its generalizations. Our algorithms are based on a novel combination of a variety of ideas, including a carefully designed probability distribution that exploits sparse coverage functions. These algorithms substantially generalize the results in Jain et al. [SODA 2023] for CC-MaxSAT and MaxCov for $K_{d,d}$-free set systems (i.e., no $d$ sets share $d$ elements), as well as a recent FPT-AS for Matroid-Constrained MaxCov by Sellier [ESA 2023] for frequency-$d$ set systems.<|reference_end|>
@article{inamdar2024satisfiability,
    title={Satisfiability to Coverage in Presence of Fairness, Matroid, and Global
  Constraints},
    author={Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket
  Saurabh, Anannya Upasana},
    journal={arXiv preprint arXiv:2403.07328},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.07328},
    primaryClass={cs.DS}
}