Parallel and Distributed Systems Report Series Complexity of Coordinating Autonomous Planning Agents Information about Parallel and Distributed Systems Report Series

We consider the problem of coordinating autonomous agents that have to perform a joint task. This joint task consists of a set of elementary tasks, partially ordered by a set of precedence constraints. Each agent is assigned (using some given task allocation protocol) a subset of the available tasks. We assume that agents wish to be fully independent during the planning process, yet they are dependent on each other because of the precedences between tasks allocated to different agents. In this paper we concentrate on the following coordination problem: how to offer complete autonomy for the participating agents in planning their part of the tasks, while at the same time guaranteeing that the individually constructed feasible plans (whatever they may be) are fully respected in constructing the joint plan. As we will see, this problem comes down to finding a minimal set of additional constraints (a coordination set) such that after adding these constraints a combined plan for executing all tasks can be achieved by simply joining whatever plans have been developed by the individual agents. We show that this problem can be decomposed into two subproblems: verifying that a given set of additional constraints is a coordination set, i.e., allows agents to plan independently (the coordination verification problem), and finding out whether a coordination set is of minimal size (the minimal coordination problem). We will show that the coordination verification problem alone is co-NPcomplete. In the general case, if the only dependencies are precedence constraints between tasks, the coordination problem is Σp2-complete; if we also allow simultaneity constraints between tasks (i.e., constraints specifying that tasks may not be performed simultaneously), then the coordination problem is Πp3-complete. Even rather simple cases of the coordination problem turn out to be intractable and we show that it is very unlikely that constant-ratio approximation algorithms for this problem exist even if each agent has only a trivial planning task to perform.