Understanding model counting for β\beta-acyclic CNF-formulas

We extend the knowledge about so-called structural restrictions of $\mathrm{\#SAT}$ by giving a polynomial time algorithm for $\beta$-acyclic $\mathrm{\#SAT}$. In contrast to previous algorithms in the area, our algorithm does not proceed by dynamic programming but works along an elimination order, solving a weighted version of constraint satisfaction. Moreover, we give evidence that this deviation from more standard algorithm is not a coincidence, but that there is likely no dynamic programming algorithm of the usual style for $\beta$-acyclic $\mathrm{\#SAT}$

Chordality properties on graphs and minimal conceptual connections in semantic data models

AbstractIn this paper the problem of finding a minimal connection among a set of objects that represent conceptual entities in a semantic data model is investigated. If we represent the conceptual structure of reality by means of a graph this problem corresponds to finding a Steiner tree over a given set of nodes. In this paper the case of bipartite graphs is considered and it is shown that, if the bipartite graphs satisfy suitable chordality properties, the Steiner problem may be solved in polynomial time. Furthermore, it is shown that such chordality properties correspond to the concepts of acyclicity that are usually considered in the relational model of data