Linear comparison for partially log-weighted incidence treewidth

Determine whether the partially log-weighted incidence treewidth of a CNF formula admits an upper bound linear in the parameter obtained by minimizing partially log-weighted width over arbitrary rooted tree decompositions and clause-path families.

Background

The paper defines partially log-weighted incidence treewidth by minimizing over nice tree decompositions and clause-path families. The remark compares this parameter with a quantity w defined by minimizing the analogous partially log-weighted width over all rooted tree decompositions and clause-path families.

The authors prove only the quadratic bound tw_plog*(F) ≤ (w+1)^2−1. A linear comparison would improve the relationship between the two formulations and could sharpen the resulting resolution-refutation bounds, although the quadratic comparison already yields an FPT bound parameterized by w.