Coalition formation is a fundamental form of interaction that allows 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 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. Now since the number of possible coalitions grows exponentially with the number of agents involved, then, instead of having a single agent calculate all these values, it would be more efficient to distribute this calculation among all agents, thus, exploiting all computational resources that are available to the system, and preventing the existence of a single point of failure. Against this background, we develop a novel algorithm for distributing the value calculation among the cooperative agents. Specifically, by using our algorithm, each agent is assigned some part of the calculation such that the agents' shares are exhaustive and disjoint. Moreover, the algorithm is decentralized, requires no communication between the agents, has minimal memory requirements, and can reflect variations in the computational speeds of the agents. To evaluate the effectiveness of our algorithm we compare it with the only other algorithm available in the literature for distributing the coalitional value calculations (due to Shehory and Kraus). This shows that for the case of 25 agents, the distribution process of our algorithm took less than 0.02% of the time, the values were calculated using 0.000006% of the memory, the calculation redundancy was reduced from 383229848 to 0, and the total number of bytes sent between the agents dropped from 1146989648 to 0. Note that for larger numbers of agents, these improvements become exponentially better. Once the coalitional 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. This problem, which is widely known as the coalition structure generation problem, is extremely challenging due to the number of possible combinations which grows very quickly as the number of agents increases, making it impossible to go through the entire search space, even for small numbers of agents. Given this, 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 a novel, anytime algorithm for solving the coalition structure generation problem. Specifically, the algorithm can generate solutions by partitioning the space of all potential coalition structures into sub-spaces containing coalition structures that are similar, according to some criterion, such that these sub-spaces can be pruned by identifying their bounds. Using this representation, the algorithm can then search through the selected sub-space(s) very efficiently using a branch-and-bound technique. We empirically show that we are able to find solutions that are optimal in 0.082% of the time required by the fastest available algorithm in the literature (for 27 agents), and that is using only 33% of the memory required by that algorithm. Moreover, our algorithm is the first to be able to solve the coalition structure generation problem for numbers of agents bigger than 27 in reasonable time (less than 90 minutes for 30 agents as opposed to around 2 months for the current state of the art). 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, given 21 agents, and after only 0.0000002% of the search space has been searched, our algorithm usually guarantees that the solution quality is no worse than 91% of optimal value, while previous algorithms only guarantees 9.52%. Moreover, our guarantee usually reaches 100% after 0.0000019% of the space has been searched, while the guarantee provided by other algorithms can never go beyond 50% until the whole space has been searched. Again note that these improvements become exponentially better given larger numbers of agents.
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