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We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain "obstruction graph" that we associate with<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00236-007-0056-x">doi:10.1007/s00236-007-0056-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/onhqc2ooj5fxjduv4ubachwn6e">fatcat:onhqc2ooj5fxjduv4ubachwn6e</a> </span>
more »... e given formula. This approach gives rise to a new hardness index for formulas, the clustering-width. Our algorithm runs in uniform polynomial time on formulas with bounded clustering-width. It is known that the number of models of formulas with bounded cliquewidth, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clustering-width and the other parameters mentioned are incomparable: there are formulas with bounded clustering-width and arbitrarily large clique-width, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clusteringwidth and bounded clique-width, treewidth, and branchwidth.
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