Planning as Branch and Bound: A Constraint Programming Implementation

Branching and lower bounds are two key notions in heuristic search and combinatorial optimization. Branching refers to the way the space of solutions is searched, while lower bounds refer to approximate cost measures used for focusing and pruning the search. In AI Planning, admissible heuristics or lower bounds have received considerable attention and most current optimal planners use them, either explicitly or implicitly (e.g., by relying on a plan graph). Branching, on the other hand, has received less attention and is seldom discussed explicitly. In this paper, we make the notion of branching in planning explicit and relate it to branching schemes used in combinatorial optimization. We analyze from this perspective the relationship between heuristic-search, constraint-based and partial-order planning, and between planning and scheduling. We also introduce a branch-and-bound formulation for planning that handles actions with durations and use unary resources, and consider a range of theories that stand halfway between planning and scheduling. The goals are twofold: to make sense of various techniques that have been found useful in planning and scheduling in a unified framework, and to lay the ground for systems that can effectively combine both capabilities. We have also implemented a planner based on this formulation on top of a constraint programming language and present some preliminary results.