A Negative Conjunctive Query is Easy if and only if it is Beta-Acyclic
Résumé
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 con- junctive 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 sat- isfiability 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(n2 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.
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