Time management under uncertainty is essential to large scale projects. From space exploration to industrial production, there is a need to schedule and perform activities. given complex specifications on timing. In order to generate schedules that are robust to uncertainty in the duration of activities, prior work has focused on a problem framing that uses an interval-bounded uncertainty representation. However, such approaches are unable to take advantage of known probability distributions over duration. In this paper we concentrate on a probabilistic formulation of temporal problems with uncertain duration, called the probabilistic simple temporal problem. As distributions often have an unbounded range of outcomes, we consider chance-constrained solutions, with guarantees on the probability of meeting temporal constraints. By considering distributions over uncertain duration, we are able to use risk as a resource, reason over the relative likelihood of outcomes, and derive higher utility solutions. We first demonstrate our approach by encoding the problem as a convex program. We then develop a more efficient hybrid algorithm whose parent solver generates risk allocations and whose child solver generates schedules for a particular risk allocation. The child is made efficient by leveraging existing interval-bounded scheduling algorithms, while the parent is made efficient by extracting conflicts over risk allocations. We perform numerical experiments to show the advantages of reasoning over probabilistic uncertainty, by comparing the utility of schedules generated with risk allocation against those generated from reasoning over bounded uncertainty. We also empirically show that solution time is greatly reduced by incorporating conflict-directed risk allocation.

In this paper, we define and study the general framework of Simple Temporal Problems with Taboo regions (STPTs) and show how these problems capture metric temporal reasoning aspects which are common to many real-world applications. STPTs encode simple temporal constraints between events and user-defined taboo regions on the timeline, during which no event is allowed to take place. We discuss two different variants of STPTs. The first one deals with (instantaneous) events, while the second one allows for (durative) processes. We also provide polynomial-time algorithms for solving them. If all events or processes cannot be scheduled outside of the taboo regions, one needs to define and reason about "soft" STPTs. We show that even "soft" STPTs can be solved in polynomial time, using reductions to max-flow problems. The resulting algorithms allow for incremental computations, which is important for the successful application of our approach in real-time domains.

Many real-world applications require the successful combination of spatial and temporal reasoning. In this paper, we study the general framework of the Traveling Salesman Problem with Simple Temporal Constraints. Representationally, this framework subsumes the Traveling Salesman Problem, Simple Temporal Problems, as well as many of the frameworks described in the literature. We analyze the theoretical properties of the combined problem providing strong inapproximability results for the general problem, and positive results for some special cases.