On Restricted Disjunctive Temporal Problems: Faster Algorithms and Tractability Frontier

In 2005 T.K.S. Kumar studied the Restricted Disjunctive Temporal Problem (RDTP), a restricted but very expressive class of Disjunctive Temporal Problems (DTPs). An RDTP comes with a finite set of temporal variables, and a finite set of temporal constraints each of which can be either one of the following three types: (t_1) two-variable linear-difference simple constraint; (t_2) single-variable disjunction of many interval constraints; (t_3) two-variable disjunction of two interval constraints only. Kumar showed that RDTPs are solvable in deterministic strongly polynomial time by reducing them to the Connected Row-Convex (CRC) constraints satisfaction problem, also devising a faster randomized algorithm. Instead, the most general form of DTPs allows for multi-variable disjunctions of many interval constraints and it is NP-complete. This work offers a deeper comprehension on the tractability of RDTPs, leading to an elementary deterministic strongly polynomial time algorithm for them, significantly improving the asymptotic running times of all the previous deterministic and randomized solutions. The result is obtained by reducing RDTPs to the Single-Source Shortest Paths (SSSP) and the 2-SAT problem (jointly), instead of reducing to CRCs. In passing, we obtain a faster (quadratic time) algorithm for RDTPs having only {t_1, t_2}-constraints and no t_3-constraint. As a second main contribution, we study the tractability frontier of solving RDTPs blended with Hyper Temporal Networks (HyTNs), a disjunctive strict generalization of Simple Temporal Networks (STNs) based on hypergraphs: we prove that solving temporal problems having only t_2-constraints and either only multi-tail or only multi-head hyperarc-constraints lies in NP cap co-NP and admits deterministic pseudo-polynomial time algorithms; on the other hand, problems having only t_3-constraints and either only multi-tail or only multi-head hyperarc-constraints turns out strongly NP-complete

Decomposing quantified conjunctive (or disjunctive) formulas

Model checking---deciding if a logical sentence holds on a structure---is a basic computational task that is well known to be intractable in general. For first-order logic on finite structures, it is PSPACE-complete, and the natural evaluation algorithm exhibits exponential dependence on the formula. We study model checking on the quantified conjunctive fragment of first-order logic, namely, prenex sentences having a purely conjunctive quantifier-free part. Following a number of works, we associate a graph to the quantifier-free part; each sentence then induces a prefixed graph, a quantifier prefix paired with a graph on its variables. We give a comprehensive classification of the sets of prefixed graphs on which model checking is tractable based on a novel generalization of treewidth that generalizes and places into a unified framework a number of existing results

Reasoning about temporal relations : the maximal tractable subalgebras of Allen's interval algebra

Allen's interval algebra is one of the best established formalisms for temporal reasoning. This article provides the final step in the classification of complexity for satisfiability problems over constraints expressed in this algebra. When the constraints are chosen from the full Allen's algebra, this form of satisfiability problem is known to be NP-complete. However, eighteen tractable subalgebras have previously been identified; we show here that these subalgebras include all possible tractable subsets of Allen's algebra. In other words, we show that this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NP-complete. We obtain this dichotomy result by giving a new uniform description of the known maximal tractable subalgebras, and then systematically using a general algebraic technique for identifying maximal subalgebras with a given property