Dynamic Consistency of Conditional Simple Temporal Networks via Mean Payoff Games: a Singly-Exponential Time DC-Checking

Conditional Simple Temporal Network (CSTN) is a constraint-based graph-formalism for conditional temporal planning. It offers a more flexible formalism than the equivalent CSTP model of Tsamardinos, Vidal and Pollack, from which it was derived mainly as a sound formalization. Three notions of consistency arise for CSTNs and CSTPs: weak, strong, and dynamic. Dynamic consistency is the most interesting notion, but it is also the most challenging and it was conjectured to be hard to assess. Tsamardinos, Vidal and Pollack gave a doubly-exponential time algorithm for deciding whether a CSTN is dynamically-consistent and to produce, in the positive case, a dynamic execution strategy of exponential size. In the present work we offer a proof that deciding whether a CSTN is dynamically-consistent is coNP-hard and provide the first singly-exponential time algorithm for this problem, also producing a dynamic execution strategy whenever the input CSTN is dynamically-consistent. The algorithm is based on a novel connection with Mean Payoff Games, a family of two-player combinatorial games on graphs well known for having applications in model-checking and formal verification. The presentation of such connection is mediated by the Hyper Temporal Network model, a tractable generalization of Simple Temporal Networks whose consistency checking is equivalent to determining Mean Payoff Games. In order to analyze the algorithm we introduce a refined notion of dynamic-consistency, named \epsilon-dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction time \hat{\varepsilon} where the CSTN transits from being, to not being, dynamically-consistent. The proof technique introduced in this analysis of \hat{\varepsilon} is applicable more in general when dealing with linear difference constraints which include strict inequalities

Checking Dynamic Consistency of Conditional Hyper Temporal Networks via Mean Payoff Games (Hardness and (pseudo) Singly-Exponential Time Algorithm)

In this work we introduce the \emph{Conditional Hyper Temporal Network (CHyTN)} model, which is a natural extension and generalization of both the \CSTN and the \HTN model. Our contribution goes as follows. We show that deciding whether a given \CSTN or CHyTN is dynamically consistent is \coNP-hard. Then, we offer a proof that deciding whether a given CHyTN is dynamically consistent is \PSPACE-hard, provided that the input instances are allowed to include both multi-head and multi-tail hyperarcs. In light of this, we continue our study by focusing on CHyTNs that allow only multi-head or only multi-tail hyperarcs, and we offer the first deterministic (pseudo) singly-exponential time algorithm for the problem of checking the dynamic-consistency of such CHyTNs, also producing a dynamic execution strategy whenever the input CHyTN is dynamically consistent. Since \CSTN{s} are a special case of CHyTNs, this provides as a byproduct the first sound-and-complete (pseudo) singly-exponential time algorithm for checking dynamic-consistency in CSTNs. The proposed algorithm is based on a novel connection between CSTN{s}/CHyTN{s} and Mean Payoff Games. The presentation of the connection between \CSTN{s}/CHyTNs and \MPG{s} is mediated by the \HTN model. In order to analyze the algorithm, we introduce a refined notion of dynamic-consistency, named $\epsilon$-dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction time $\hat{\varepsilon}$ where a \CSTN/CHyTN transits from being, to not being, dynamically consistent. The proof technique introduced in this analysis of $\hat{\varepsilon}$ is applicable more generally when dealing with linear difference constraints which include strict inequalities.Comment: arXiv admin note: text overlap with arXiv:1505.0082

Modeling chloride circulation at Plynlimon catchments, Wales

A recent stochastic formulation of transport processes within catchments is applied to a small catchment within Plynlimon area, Walesope

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

Incorporating Decision Nodes into Conditional Simple Temporal Networks

A Conditional Simple Temporal Network (CSTN) augments a Simple Temporal Network (STN) to include special time-points, called observation time-points. In a CSTN, the agent executing the network controls the execution of every time-point. However, each observation time-point has a unique propositional letter associated with it and, when the agent executes that time-point, the environment assigns a truth value to the corresponding letter. Thus, the agent observes but, does not control the assignment of truth values. A CSTN is dynamically consistent (DC) if there exists a strategy for executing its time-points such that all relevant constraints will be satisfied no matter which truth values the environment assigns to the propositional letters. Alternatively, in a Labeled Simple Temporal Network (Labeled STN) - also called a Temporal Plan with Choice - the agent executing the network controls the assignment of values to the so-called choice variables. Furthermore, the agent can make those assignments at any time. For this reason, a Labeled STN is equivalent to a Disjunctive Temporal Network. This paper incorporates both of the above extensions by augmenting a CSTN to include not only observation time-points but also decision time-points. A decision time-point is like an observation time-point in that it has an associated propositional letter whose value is determined when the decision time-point is executed. It differs in that the agent - not the environment - selects that value. The resulting network is called a CSTN with Decisions (CSTND). This paper shows that a CSTND generalizes both CSTNs and Labeled STNs, and proves that the problem of determining whether any given CSTND is dynamically consistent is PSPACE-complete. It also presents algorithms that address two sub-classes of CSTNDs: (1) those that contain only decision time-points; and (2) those in which all decisions are made before execution begins