Improved FPT approximation scheme and approximate kernel for biclique-free max k-weight SAT: Greedy strikes back

Pasin Manurangsi

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. [20] gave an FPT approximation scheme (FPT-AS) with running time 2O((dk/ϵ)d)⋅(n+m)O(1)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo stretchy="true" is="true">(</mo><msup is="true"><mrow is="true"><mo stretchy="true" is="true">(</mo><mi is="true">d</mi><mi is="true">k</mi><mo stretchy="false" is="true">/</mo><mi is="true">ϵ</mi><mo stretchy="true" is="true">)</mo></mrow><mrow is="true"><mi is="true">d</mi></mrow></msup><mo stretchy="true" is="true">)</mo></mrow></mrow></msup><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">m</mi><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> for Max k-Weight SAT when the incidence graph is Kd,d<math><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mi is="true">d</mi><mo is="true">,</mo><mi is="true">d</mi></mrow></msub></math>-free. They asked whether a polynomial-size approximate kernel exists. In this work, we answer this question positively by giving a (1−ϵ)<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi><mo stretchy="false" is="true">)</mo></math>-approximate kernel with (dkϵ)O(d)<math><msup is="true"><mrow is="true"><mo stretchy="true" is="true">(</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi><mi is="true">k</mi></mrow><mrow is="true"><mi is="true">ϵ</mi></mrow></mfrac><mo stretchy="true" is="true">)</mo></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo stretchy="false" is="true">)</mo></mrow></msup></math> variables. This also implies an improved FPT-AS with running time (dk/ϵ)O(dk)⋅(n+m)O(1)<math><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mi is="true">k</mi><mo stretchy="false" is="true">/</mo><mi is="true">ϵ</mi><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">m</mi><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math>. Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.