() , for reals (a single-variable disjunction of many interval-constraints);

() , for reals (a 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 problem; plus, he devised a randomized algorithm whose expected running time is less than that of the deterministic one. 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 solving them,

significantly improving the asymptotic running times of all the previous deterministic and randomized algorithms. 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 and -constraints (and no -constraint).

As a second main contribution, we study the tractability frontier of solving RDTPs blended with Hyper Temporal Networks (HyTNs), a strict generalization of Simple Temporal Networks (STNs) grounded on hypergraphs: we prove that solving temporal problems having only -constraints and either only multi-tail or only multi-head hyperarc-constraints lies in and it admits deterministic pseudo-polynomial time algorithms; on the other hand, problems having only -constraints and either only multi-tail or only multi-head hyperarc-constraints are strongly NP-complete.

Download: manuscript.

]]>Download: EGS-v0.1.

]]>**Incorporating Decision Nodes into Conditional Simple Temporal Networks**

has been accepted at TIME 2017, to be held at Université de Mons, Belgium, Oct. 2017.

Abstract. A Conditional Simple Temporal Network (CSTN) is an extension of Simple Temporal Network (STN) having some special time-points, named observation time-points. In a STN/CSTN, the agent executing the network controls the execution of every time-point. Observation time-points have unique propositional letters associated with them and, when the agent executes one of them, the environment assigns a truth value to the corresponding propositional 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 completely controls the process of assigning 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. A generalisation of the Bellman-Ford algorithm to accommodate the propagation of labeled values has been used to determine whether any given Labeled STN is consistent.

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. It proves that the decision problem of determining whether or not any CSTND is dynamically consistent is PSPACE-complete. And it presents algorithms that restrict attention to two special classes of CSTNDs: (i) those that contain only decision time-points; and (ii) those in which all decisions are made before starting to execute the network.

Download: paper.

]]>*Complexity in Infinite Games on Graphs and Temporal Constraint Networks,*

has been defended on 20 March 2017 (University of Trento and Université Paris-Est Marne la Vallée).

Download: manuscript, presentation.

]]>Download: manuscript.

]]>For instance, consider the following communication network problem. Suppose we have data stored on each node of a network and we want to continuously update all nodes with consistent data: often one requirement is to share key information between all nodes of a network; this can be done by having a data packet of current information continuously going through all nodes. Unfortunately not all routing choices are always under our control, some of them may be controlled by the network environment, which could play against us. This is essentially an infinite 2-player pebble game played on an *arena*, i.e. a finite directed simple graph in which the vertices are divided into two classes, and , where Player wants to visit all vertices infinitely often by moving the pebble on them, while Player works against. This is called *Update Game (UG)* in [Dinneen and Khoussainov (1999)].

In this work, we introduce and study a refined notion of reachability for arenas, named *trap-reachability*, where Player attempts to reach vertices without leaving a prescribed subset , while Player works against. It is shown that every arena decomposes into *strongly-trap-connected components (STCCs)*.

Our main result is a linear time algorithm for computing this unique decomposition.

Both the graph structures and the algorithm generalize the classical decomposition of a directed graph into its strongly-connected components (SCCs). The algorithm builds on a generalization of the depth-first search (DFS), taking inspiration from Tarjan’s SCCs classical algorithm. The structures of palm-trees and jungles described in Tarjan’s original paper need to be revisited and generalized (i.e. tr-palm-trees and tr-jungles) in order to handle the 2-player infinite pebble game’s setting.

This theory has direct applications in solving Update Games (UGs) faster. Dinneen and Khoussainov showed in 1999 that deciding who’s the winner in a given UG costs time, where is the number of vertices and is that of arcs. We solve that problem in linear time. The result is obtained by observing that the UG is a win for Player if and only if the arena comprises one single STCC. It is also observed that the tr-palm-tree returned by the algorithm encodes routing information that an -space agent can consult to win the UG in time per move. With this, the polynomial-time complexity for deciding Explicit McNaughton-Muller Games is also improved, from cubic to quadratic.

Download: manuscript, presentation.

]]>,

where counts the number of times that a certain energy-lifting operator is applied to any , along a certain sequence of Value-Iterations on reweighted EGs; and is the degree of . This improves significantly over a previously known pseudo-polynomial time estimate, i.e. , as the pseudo-polynomiality is now confined to depend solely on . Secondly, we further explore on the relationship between Optimal Positional Strategies (OPSs) in MPGs and Small Energy-Progress Measures (SEPMs) in reweighted EGs. It is observed that the space of all OPSs, , admits a unique complete decomposition in terms of extremal-SEPMs in reweighted EGs. This points out what we called the “Energy-Lattice associated to “. Finally, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of , and for computing the corresponding partitioning of .

Download: manuscript.

]]>TIME2016 was held at Technical University of Denmark (DTU), Copenhagen, Denmark (October 17-18-19, 2016).

This is a joint work with M. Cairo and R. Rizzi.

Pre-print link: paper, presentation.

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