Over-subscription, that is, being assigned too many tasks or requirements that are too demanding, is commonly encountered in temporal planning problems. As human beings, we often want to do more than we can, ask for things that may not be available, while underestimating how long it takes to perform each task. It is often difficult for us to detect the causes of failure in such situations and then find resolutions that are effective. We can greatly benefit from tools that assist us by looking out for these plan failures, by identifying their root causes, and by proposing preferred resolutions to these failures that lead to feasible plans. In recent literature, several approaches have been developed to resolve such over-subscribed problems, which are often framed as over-constrained scheduling, configuration design or optimal planning problems. Most of them take an all-or-nothing approach, in which over-subscription is resolved through suspending constraints or dropping goals. While helpful, in real-world scenarios, we often want to preserve our plan goals as much possible. As human beings, we know that slightly weakening the requirements of a travel plan, or replacing one of its destinations with an alternative one is often sufficient to resolve an over-subscription problem, no matter if the requirement being weakened is the duration of a deep-sea survey being planned for, or the restaurant cuisine for a dinner date. The goal of this thesis is to develop domain independent relaxation algorithms that perform this type of slight weakening of constraints, which we will formalize as continuous relaxation, and to embody them in a computational aid, Uhura, that performs tasks akin to an experienced travel agent or ocean scientists. In over-subscribed situations, Uhura helps us diagnose the causes of failure, suggests alternative plans, and collaborates with us in order to resolve conflicting requirements in the most preferred way. Most importantly, the algorithms underlying Uhura supports the weakening, instead of suspending, of constraints and variable domains in a temporally flexible plan. The contribution of this thesis is two-fold. First, we developed an algorithmic framework, called Best-first Conflict-Directed Relaxation (BCDR), for performing plan relaxation. Second, we use the BCDR framework to perform relaxation for several different families of plan representations involving different types of constraints. These include temporal constraints, chance constraints and variable domain constraints, and we incorporate several specialized conflict detection and resolution algorithms in support of the continuous weakening of them. The key idea behind BCDR's approach to continuous relaxation is to generalize the concepts of discrete conflicts and relaxations, first introduced by the model-based diagnosis community, to hybrid conflicts and relaxations, which denote minimal inconsistencies and minimal relaxations to both discrete and continuous relaxable constraints.
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