Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints
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2024
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| author | Inamdar, Tanmay Jain, Pallavi Lokshtanov, Daniel Sahu, Abhishek Saurabh, Saket Upasana, Anannya |
| author_facet | Inamdar, Tanmay Jain, Pallavi Lokshtanov, Daniel Sahu, Abhishek Saurabh, Saket Upasana, Anannya |
| contents | In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $Φ$, and $k \ge 0$, and the goal is to find an assignment $β$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $β$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $Φ$ 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 $O(1/ε)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-ε$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints Inamdar, Tanmay Jain, Pallavi Lokshtanov, Daniel Sahu, Abhishek Saurabh, Saket Upasana, Anannya Data Structures and Algorithms In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $Φ$, and $k \ge 0$, and the goal is to find an assignment $β$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $β$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $Φ$ 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 $O(1/ε)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-ε$. 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. |
| title | Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2403.07328 |