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Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints

Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket Saurabh, Anannya Upasana

TL;DR

A randomized reduction is given 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 $1-\epsilon$.

Abstract

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.

Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints

TL;DR

A randomized reduction is given from CC-MaxSAT to MaxCov in time that preserves the approximation guarantee up to a factor of .

Abstract

In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula , and , and the goal is to find an assignment with at most variables set to true (also called a weight -assignment) such that the number of clauses satisfied by is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula 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 (approximation is in terms of number of clauses satisfied/elements covered). We give a randomized reduction from CC-MaxSAT to MaxCov in time that preserves the approximation guarantee up to a factor of . 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 -free set systems (i.e., no sets share elements), as well as a recent FPT-AS for Matroid-Constrained MaxCov by Sellier [ESA 2023] for frequency- set systems.
Paper Structure (28 sections, 21 theorems, 34 equations, 1 figure, 3 algorithms)

This paper contains 28 sections, 21 theorems, 34 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Tractability Boundaries for Maximum Coverage
  3. Matroid Constraints
  4. Fairness or Multiple Coverage Constraints
  5. Our New Problem: Combining Matroid and Fairness Constraints
  6. Related Results
  7. Overview of Our Results and Techniques
  8. Reduction from CC-MaxSat to Maximum Coverage: An overview of \ref{['thmintro:reduction']}
  9. Deterministic and Randomized Branching using a Largest Set
  10. Insight into the probabilistic branching.
  11. Handling fairness constraints via the Bucketing trick
  12. Handling Matroid Constraints.
  13. Further Extensions
  14. $K_{d,d}$-free Set Systems.
  15. Handling Multiple Matroid Constraints.
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\epsilon >0$. There is a polynomial time randomized algorithm that given an instance $(\Phi, k)$ of CC-MaxSat produces an instance $(U, {\cal F}, k)$ of Maximum Coverage such that the following holds with probability $\frac{1}{2}(\frac{\epsilon}{2})^{k}$. Given a $(1-\epsilon) {\sf OPT_{cov}}$

Figures (1)

  • Figure 1: If there is an arrow of the form $A\rightarrow B$, then problem $B$ generalizes problem $A$. FPT-ASes for the problems in red bubbles are not known in the literature, and we study in this paper. For all the other problems FPT-ASes are known in the literature for some cases. This paper improves the results in cyan and blue.

Theorems & Definitions (39)

  • Theorem 1.1: Informal
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 3.1: fomin2014efficientDBLP:books/sp/CyganFKLMPPS15
  • Proposition 3.2: fomin2014efficientDBLP:books/sp/CyganFKLMPPS15
  • Lemma 3.3
  • proof
  • ...and 29 more