An Anytime Algorithm for Optimal Coalition Structure Generation (1401.3466v1)

Abstract: Coalition formation is a fundamental type of interaction that involves the creation of coherent groupings of distinct, autonomous, agents in order to efficiently achieve their individual or collective goals. Forming effective coalitions is a major research challenge in the field of multi-agent systems. Central to this endeavour is the problem of determining which of the many possible coalitions to form in order to achieve some goal. This usually requires calculating a value for every possible coalition, known as the coalition value, which indicates how beneficial that coalition would be if it was formed. Once these values are calculated, the agents usually need to find a combination of coalitions, in which every agent belongs to exactly one coalition, and by which the overall outcome of the system is maximized. However, this coalition structure generation problem is extremely challenging due to the number of possible solutions that need to be examined, which grows exponentially with the number of agents involved. To date, therefore, many algorithms have been proposed to solve this problem using different techniques ranging from dynamic programming, to integer programming, to stochastic search all of which suffer from major limitations relating to execution time, solution quality, and memory requirements. With this in mind, we develop an anytime algorithm to solve the coalition structure generation problem. Specifically, the algorithm uses a novel representation of the search space, which partitions the space of possible solutions into sub-spaces such that it is possible to compute upper and lower bounds on the values of the best coalition structures in them. These bounds are then used to identify the sub-spaces that have no potential of containing the optimal solution so that they can be pruned. The algorithm, then, searches through the remaining sub-spaces very efficiently using a branch-and-bound technique to avoid examining all the solutions within the searched subspace(s). In this setting, we prove that our algorithm enumerates all coalition structures efficiently by avoiding redundant and invalid solutions automatically. Moreover, in order to effectively test our algorithm we develop a new type of input distribution which allows us to generate more reliable benchmarks compared to the input distributions previously used in the field. Given this new distribution, we show that for 27 agents our algorithm is able to find solutions that are optimal in 0.175% of the time required by the fastest available algorithm in the literature. The algorithm is anytime, and if interrupted before it would have normally terminated, it can still provide a solution that is guaranteed to be within a bound from the optimal one. Moreover, the guarantees we provide on the quality of the solution are significantly better than those provided by the previous state of the art algorithms designed for this purpose. For example, for the worst case distribution given 25 agents, our algorithm is able to find a 90% efficient solution in around 10% of time it takes to find the optimal solution.

• The paper introduces an anytime algorithm that efficiently partitions the search space to compute optimal coalition structures in NP-complete settings.
• It employs a novel cyclation technique with a branch-and-bound strategy, avoiding redundant evaluations and significantly accelerating the search process.
• Empirical results demonstrate that the algorithm finds optimal solutions in just 0.175% of the time required by previous methods, proving its effectiveness in dynamic environments.

An Anytime Algorithm for Optimal Coalition Structure Generation

The research paper entitled "An Anytime Algorithm for Optimal Coalition Structure Generation" presents a significant advancement in the domain of multi-agent systems (MAS), specifically focusing on the coalition structure generation (CSG) problem. The paper addresses the inherent complexity of forming optimal coalitions—an NP-complete problem—by introducing a novel anytime algorithm that enhances both the efficiency and quality of solutions.

The authors, Rahwan et al., propose an innovative approach to partition the search space into sub-spaces based on integer partitions of the number of agents. This new partitioning facilitates efficient computation of upper and lower bounds on coalition values, enabling significant pruning of potential solutions that cannot improve upon the current best coalition structure. This representation not only reduces the complexity of the search within these sub-spaces but allows for a more informed trade-off analysis between coalition sizes and solution quality, effectively bypassing the exponential blow-up associated with the whole search space.

The algorithm employs a novel cyclation technique, avoiding redundant and invalid coalition structures. It does not repetitively explore the same coalition structures, ensuring each is considered only once, thus improving computational efficiency. Coupled with a branch-and-bound methodology, the algorithm efficiently zooms in on potentially better solutions, applying bounds dynamically to further reduce unnecessary computations within each sub-space.

Performance evaluation indicates that IP, the proposed algorithm, significantly outperforms prior methods, notably the Improved Dynamic Programming (IDP) approach. Specifically, in certain scenarios, IP finds optimal solutions in just 0.175% of the time required by IDP when considering 27 agents. Such performance not only highlights the robustness of IP in generating optimal solutions expeditiously but also underscores its ability to deliver high-quality suboptimal solutions rapidly if interrupted—ensuring anytime characteristics. In empirical tests, IP consistently outperformed Sandholm et al.'s and Dang and Jennings’s methods by providing better solution guarantees quickly.

The consideration of different input distributions, notably the NDCS, elucidates the lack of bias in the search space, in contrast to previous Normal and Uniform distributions. Such thoughtful design ensures that the bounding characteristics employed in IP are neither unduly optimistic nor pessimistic, improving the practical applicability of the algorithm across varied operational scenarios within multi-agent systems.

In a practical sense, IP's benefits are manifold. The ability to compute near-optimal solutions quickly is invaluable in dynamic environments where time constraints are prevalent, and decision making cannot always await a fully optimal solution. Moreover, the flexibility to incorporate preferences for coalition sizes allows agents to adapt the search to specific contextual needs, further enhancing the algorithm's utility.

Theoretically, this work offers a solid foundation for further exploration into combinatorial optimization problems akin to coalition structure generation, providing a direction for how the underlying mathematical principles can be extended to similar domains like combinatorial auctions or crew scheduling.

In conclusion, the paper advances the state-of-the-art in coalition structure generation by presenting an efficient, scalable, and practical solution aligned with the intricate demands of today's multi-agent systems. Future developments could extend its applicability and optimize its computational footprint even further, suggesting an exciting trajectory for subsequent research in this rapidly evolving field.