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 $\mathrm{\#SAT}$

A Negative Conjunctive Query is Easy if and only if it is Beta-Acyclic

It is known that the data complexity of a Conjunctive Query (CQ) is determined only by the way its variables are shared between atoms, reflected by its hypergraph. In particular, Yannakakis [18, 3] proved that a CQ is decidable in linear time when it is α-acyclic, i.e. its hypergraph is α-acyclic; Bagan et al. [2] even state: Any CQ is decidable in linear time iff it is α-acyclic. (under certain hypotheses) (By linear time, we mean a query on a structure S can be decided in time O(|S|)) A natural question is: since the complexity of a Negative Conjunctive Query (NCQ), a conjunctive query where all atoms are negated, also only depends on its hypergraph, can we find a similar dichotomy in this case? To answer this question, we revisit a result of Ordyniak et al. [17] — that states that satisfiability of a β-acyclic CNF formula is decidable in polynomial time — by proving that some part of their procedure can be done in linear time. This implies, under an algorithmic hypothesis (precisely: one cannot decide whether a graph is triangle-free in time O(n 2 log n) where n is the number of vertices.) that is likely true: Any NCQ is decidable in quasi-linear time iff it is β-acyclic. (By quasi-linear time, we mean a query on a structure S can be decided in time O(|S | log |S|)) We extend the easiness result to Signed Conjunctive Query (SCQ) where some atoms are negated. This has great interest since using some negated atoms is natural in the frameworks of databases and CSP. Furthermore, it implies straightforwardly the following: Any β-acyclic existential first-order query is decidable in quasi-linear time

Models of Horn Formulas are Enumerable at Nearly Linear Delay

7 pages Paper submitted to journal Information Processing Letters in May 2012The Unique satisfiability problem of Horn formulas was proved to be quadratic by Minoux (1992) [3]. Using ideas based on those of [3], Berman, Franco, and Schilpf (1995) [1] proved it to be nearly linear. We simplify the presentation of their algorithm and adapt it slightly in order to perform enumeration of the solutions at a delay that is their unique solution decision time, i.e. O(\alpha(|F |)|F |) where |F| is the formula size and \alpha is the inverse Ackermann function, or (at choice) O(n log n + |F |) where n is the number of variables of F

De la pertinence de l’énumération : complexité en logiques propositionnelle et du premier ordre

Beyond the decision of satisfiability problems, we investigate the problemof enumerating all their solutions.In a first part, we consider the enumeration problem in the frameworkof the propositional satisfiability problem. Creignou and Hébrard provedthat the polynomial classes for the non-trivial sat problem are exactlythose for the enumeration problem. We give optimal enumerationalgorithms for each of these classes, that generalize any non-trivialdecision algorithm for this class. This suggests that enumeration is therelevant problem in this case, rather than the decision problem.In a second part, we simplify and complete some results of Baganet al. that establish a strong connection between the tractability of aconjunctive query and a notion of hypergraph acyclicity. We establishsimilar results for the dual class of the class of conjunctive queries,thanks to a new algorithm. Finally, we generalize all these resultsthrough a single dichotomy for the enumeration problem of conjunctivesigned queries, by generalizing some classical combinatorial result byBrouwer and Kolen. This dichotomy establishes a close connectionbetween enumeration strong tractability and decision strong tractability.Au-delà de la décision de problèmes de satisfaisabilité, on s’intéresse à la génération exhaustive de leurs solutions, l’énumération.Nous interrogeons d’abord la pertinence du problème d’énumération dans le cadre très classique de la logique propositionnelle. La dichotomie de Creignou et Hébrard prouve déjà l’équivalence entre les classes polynomiales pour la décision non triviale et celles pour l’énumération. On donne des algorithmes d’énumération optimaux pour chacune de ces classes, qui généralisent tout algorithme de décision non triviale, suggérant que l’énumération est le problème pertinent dans ce cadre.Ensuite, nous complétons et simplifions des résultats de dichotomie de Bagan et al. qui établissent un lien étroit entre la facilité d’une requête conjonctive et une notion d’acyclicité d’hypergraphe. On prouve alors, grâce à un nouvel algorithme, des résultats similaires pour la classe duale de celle des requêtes conjonctives. Finalement, en généralisant le résultat classique de combinatoire de Brouwer et Kolen, on unifie l’ensemble de ces résultats sous forme d’une dichotomie pour l’énumération des requêtes conjonctives dites signées, qui établit un lien fort entre facilité de l’énumération et facilité de la décision