Section: New Results

Fundamental results and algorithms: distributed planning

Participants : Eric Fabre, Loig Jézéquel.

A planning problem consists in organizing some actions in order to reach an objective. Formally, this is equivalent to finding a path from an initial state to a goal/marked state in a huge automaton. The latter is specified by a collection of resources, that may be available or not (which defines a state), and actions that consume and produce resources (which defines a transition). In the case of optimal planning, actions have a cost, and the objective is to find a path of minimal cost to the goal.

Our interest in this problem is threefold. First, it is naturally an instance of a concurrent system, given that actions have local effects on resources. Secondly, it is a weak form of an optimal control problem for a concurrent/distributed system. Finally, we are interested in distributed solutions to such problems, which is an active topic in the planning community under the name of “factored planning.”

Our previous contribution to the domain was the first optimal factored planning algorithm [47] . The main idea is to represent a planning problem as a network of interacting weighted automata, the objective being to jointly drive all of them to a target state, while minimizing the cost of their joint trajectory. We have developed and tested [53] a distributed algorithm to solve this problem, based on a weighted automata calculus, and that takes the shape of a message passing procedure. Components perform local computations, exchange messages with their neighbors, in an asynchronous manner, and the procedure converges to the path that each component should follow. The optimal global plan is thus given as a tuple of (compatible) local plans, i.e. a partial order of actions.

In 2012, we have extended this framework in two directions. The first one considers large planning problems for which the interaction graph of components is not a tree. It is well known that message passing algorithms (also called belief propagation) is optimal on trees. To recover such a situation where distributed optimal planning can be resolved exactly, one therefore has to smartly group components into larger ones in order to recover a tree of larger components. This is done at the expense of the complexity in the resolution of local planning problems (which augments exponentially with the number of assembled components). Alternately, one can also ignore that the graph is not a tree, and thus use the so-called loopy belief propagation, which requires minor adaptations. This results in a new approach to the resolution of planning problems, where approximate solutions are provided: one can check that the computed plans are valid, but their optimality is not guaranteed. We have experimented this turbo-planning idea on a series of random benchmarks, some of them being not accessible to standard planning methods. The results are surprisingly good: distributed plans are found in most cases, and are often close to optimal. However, no theoretical results can yet support this phenomenon [30] .

The second extension to distributed planning concerns the multi-agent version of the central A* (A-star) algorithm, which is at the core of numerous planners. By contrast with the previous setting, we do not build all plans here, in a distributed manner, but perform a search for an optimal plan. The centralized version of A* performs a depth-first search of a winning path in a graph, guided by some heuristic function that orients the search towards the goal. In our setting, several path searches must be performed in the graphs of the different components (or local planning problems), under the constraint that the provided paths are compatible, i.e. agree on the execution of the common actions. The resulting local paths must also be jointly optimal, once their costs are added. We have proposed a complete solution to this problem, called A# (A-sharp) [29] . Our efforts now aim at mixing these ideas with the turbo planning approach.