Optimal Multi-Agent Scheduling with Constraint Programming

We consider the problem of computing optimal schedules in multi-agent systems. In these problems, actions of one agent can influence the actions of other agents, while the objective is to maximize the total ‘quality’ of the schedule. More specifically, we focus on multiagent scheduling problems with time windows, hard and soft precedence relations, and a nonlinear objective function. We show how we can model and efficiently solve these problems with constraint programming technology. Elements of our proposed method include constraint-based reasoning, search strategies, problem decomposition, scheduling algorithms, and a linear programming relaxation. We present experimental results on realistic problem instances to display the different elements of the solution process. Introduction Multi-agent planning and scheduling problems arise in many contexts such as supply chain management, coordinating space missions, or con guring and executing military scenarios. In these situations, the agents usually need to perform certain tasks in order to achieve a common goal. Often the agents need to respect various restrictions such as temporal constraints and interdependency relations. Furthermore, depending on the application at hand, these problems may be subject to several uncertainties, for example the actual outcome and duration of executing a task, and changing environmental conditions. Multi-agent planning and scheduling problems are among the most dif cult problems in Arti cial Intelligence. While the centralized deterministic version is already NP-hard, the non-deterministic distributed version is even NEXP-complete (Bernstein et al. 2002). In this paper we present an ef cient method to compute provably optimal solutions for centralized deterministic multi-agent scheduling problems. The motivation for our work stems from the application studied in the DARPA program COORDINATORs. In this application, agents correspond to military units that need to achieve a common goal. Initially, each agent has its own local view of the situation, and an initial schedule of tasks to execute. During the course of action, the actual duration and quality of executed tasks as Copyright c © 2007, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. well as environmental changes typically force the agents to adapt their schedule. Because of interdependency relations between the tasks, proposed changes must be communicated and negotiated with the other agents. The aim of the project is to automate the coordination process of negotiation and rescheduling in a distributed fashion. Our system, described in this paper, is able to compute provably optimal centralized solutions for deterministic scenarios sketched above. Within the program, our system is applied in several ways. Most importantly, it is used to evaluate the performance of distributed approaches. Namely, for a given (simulated) problem, we create a deterministic problem by replacing all uncertainty with the actual outcomes. The optimal centralized solution to this problem serves as an upper bound for the performance of a distributed approach. To obtain an average expected performance bound to a problem, we average the optimal solutions for a suf ciently large number of samples instead. Furthermore, sampling the outcome space has also been applied to evaluate the test problems themselves. Problem instances containing rare outcome outliers resulting in extremely low or high solution quality should ideally be avoided for performance evaluation. We recognize such problems by analyzing the distribution of optimal solutions over a large number of samples. Finally, our system is also in use to study adaptive algorithm selection procedures (Rosenfeld 2007), and as part of an environment to simulate user interaction (Sarne & Grosz 2007), within this program. Hence, the main requirements for our system are guaranteed optimality and computational ef ciency. As the experiments will show, we can optimally solve large problem instances involving 2250 actions and 100 agents, in only 13 seconds of computation time. Our approach is based on constraint programming technology. This has several advantages. First, it allows us to specify the problem in a rich modeling language, and to apply the corresponding default constraint-based reasoning. As we will see below, our model is very close to the original representation of the problem. Second, in the constraint programming framework we can specify detailed search heuristics, tailored to the speci c needs of the problem. In addition, we have implemented a problem decomposition scheme to further improve our search process. Third, we have implemented an “optimization constraint”, based on a linear programming relaxation of the problem, to strengthen Q: 8 D: 6 A: John Method5