FPT-sized resolution refutations parameterized by incidence treewidth

Establish whether every unsatisfiable CNF formula has an FPT-sized resolution refutation parameterized by incidence treewidth.

Background

The paper studies upper bounds on the length of resolution refutations for unsatisfiable CNF formulas in terms of incidence treewidth and two weighted variants: log-weighted incidence treewidth and partially log-weighted incidence treewidth. It proves several FPT-sized bounds for these weighted parameters, as well as bounds depending jointly on ordinary incidence treewidth and maximum clause width.

The unrestricted question for ordinary incidence treewidth remains unresolved: the results do not establish an FPT bound for all unsatisfiable CNF formulas parameterized solely by incidence treewidth, nor do they provide lower bounds ruling out such a bound. A positive resolution would require a construction covering arbitrary clause widths, while a negative resolution would require stronger lower-bound techniques.