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Improved FPT Approximation Scheme and Approximate Kernel for Biclique-Free Max k-Weight SAT: Greedy Strikes Back

Pasin Manurangsi

TL;DR

This work delivers a polynomial-size $(1-oldsymbol{ ext{ε}})$-approximate kernel for $K_{a,b}$-free Max $k$-Weight SAT, resolving an open question about kernelization for this problem class. The kernel combines three reductions—eliminating many negative literals, pruning positive variables via a sunflower-lemma argument under sparsity, and scaling/rounding clauses—resulting in a bound of $Oig( rac{k \, ext{log} k}{oldsymbol{ ext{ε}}}ig) + a \, O(b)^{2b} \, rac{k^b}{oldsymbol{ ext{ε}}^{3b}}$ variables and $(k/oldsymbol{ ext{ε}})^{O(ab)}$ clauses. Consequently, the authors obtain an improved FPT approximate scheme time of $ig( rac{dk}{oldsymbol{ ext{ε}}}ig)^{O(dk)}$ (and $ig( rac{a^{1/b} \, b \, k}{oldsymbol{ ext{ε}}}ig)^{O(bk)}$ in general), by enabling exhaustive search on the reduced instance. The approach leverages greedy reductions with a sunflower lemma specialized for $K_{a,b}$-free graphs, yielding deterministic kernelization that complements recent randomized results in the literature and broadens the toolkit for parameterized approximation of SAT-related problems.

Abstract

In the Max $k$-Weight SAT (aka Max SAT with Cardinality Constraint) problem, we are given a CNF formula with $n$ variables and $m$ clauses together with a positive integer $k$. The goal is to find an assignment where at most $k$ variables are set to one that satisfies as many constraints as possible. Recently, Jain et al. [SODA'23] gave an FPT approximation scheme (FPT-AS) with running time $2^{O\left(\left(dk/ε\right)^d\right)} \cdot (n + m)^{O(1)}$ for Max $k$-Weight SAT when the incidence graph is $K_{d,d}$-free. They asked whether a polynomial-size approximate kernel exists. In this work, we answer this question positively by giving an $(1 - ε)$-approximate kernel with $\left(\frac{d k}ε\right)^{O(d)}$ variables. This also implies an improved FPT-AS with running time $(dk/ε)^{O(dk)} \cdot (n + m)^{O(1)}$. Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.

Improved FPT Approximation Scheme and Approximate Kernel for Biclique-Free Max k-Weight SAT: Greedy Strikes Back

TL;DR

This work delivers a polynomial-size -approximate kernel for -free Max -Weight SAT, resolving an open question about kernelization for this problem class. The kernel combines three reductions—eliminating many negative literals, pruning positive variables via a sunflower-lemma argument under sparsity, and scaling/rounding clauses—resulting in a bound of variables and clauses. Consequently, the authors obtain an improved FPT approximate scheme time of (and in general), by enabling exhaustive search on the reduced instance. The approach leverages greedy reductions with a sunflower lemma specialized for -free graphs, yielding deterministic kernelization that complements recent randomized results in the literature and broadens the toolkit for parameterized approximation of SAT-related problems.

Abstract

In the Max -Weight SAT (aka Max SAT with Cardinality Constraint) problem, we are given a CNF formula with variables and clauses together with a positive integer . The goal is to find an assignment where at most variables are set to one that satisfies as many constraints as possible. Recently, Jain et al. [SODA'23] gave an FPT approximation scheme (FPT-AS) with running time for Max -Weight SAT when the incidence graph is -free. They asked whether a polynomial-size approximate kernel exists. In this work, we answer this question positively by giving an -approximate kernel with variables. This also implies an improved FPT-AS with running time . Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.
Paper Structure (16 sections, 11 theorems, 16 equations)

This paper contains 16 sections, 11 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Technical Overview.
  4. On Independent Work of Inamdar et al. IJLSSU24.
  5. Preliminaries
  6. Approximate Kernel for Max $k$-Weight SAT
  7. Step I: Reducing # Negative Variables
  8. Step II: Reducing # Positive Variables
  9. A Sunflower Lemma
  10. The Preprocessing Algorithm
  11. Case I: $\tau \geq \frac{2 b}{\epsilon}$.
  12. Case II: $\tau < \frac{2b}{\epsilon}$.
  13. Step III: Reducing # Clauses
  14. Putting Things Together: Proof of \ref{['thm:main-kernel']}
  15. Discussion and Open Questions
  16. ...and 1 more sections

Key Result

Theorem 1

For any $a, b \in \mathbb{N}$ and $\epsilon \in (0, 1/2)$, there is a parameter-preservingWe say that an approximate kernel is parameter-preserving if we have $k' = k$.$(1 - \epsilon)$-approximate kernel for $K_{a,b}$-free Max $k$-Weight SAT with $O\left(\frac{k \log k}{\epsilon}\right) + a \cdot O(

Theorems & Definitions (20)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Theorem 4: Sviridenko01
  • Lemma 5: Clause Modification APPA
  • proof
  • Lemma 6: Variable Deletion APPA
  • proof
  • Lemma 7
  • ...and 10 more