Positive andNegative Results for Parameterized Compilability

Most relevant computational tasks are not polynomial-size compilable, even though they are compiled rather efficiently in practice. Parameterized compilability offers a framework to narrow this gap between theoretical incompilability and practical compilation by relaxing the size bound posed on compilations. We study various computational problems (related to clause entailment, the classical problem of checking whether a CNF formula entails a clause) in the framework of parameterized compilability. We establish several positive and negative results. On the positive side we show, for instance, that clause entailment is fixed-parameter compilable parameterized by the minimum incidence treewidth over all equivalent CNF formulas. On the negative side we show, for instance, that the size of any compilation of the clause entailment problem restricted to clauses of bounded size has an exponential dependency on the size bound posed on the clauses.