Complexity of coalition structure generation

We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds.Comment: 17 page

Complexity of Coalition Structure Generation

We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds

Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph

Coalition structure generation in cooperative games with compact representations

This paper presents a new way of formalizing the coalition structure generation problem (CSG) so that we can apply constraint optimization techniques to it. Forming effective coalitions is a major research challenge in AI and multi-agent systems. CSG involves partitioning a set of agents into coalitions to maximize social surplus. Traditionally, the input of the CSG problem is a black-box function called a characteristic function, which takes a coalition as input and returns the value of the coalition. As a result, applying constraint optimization techniques to this problem has been infeasible. However, characteristic functions that appear in practice often can be represented concisely by a set of rules, rather than treating the function as a black box. Then we can solve the CSG problem more efficiently by directly applying constraint optimization techniques to this compact representation. We present new formalizations of the CSG problem by utilizing recently developed compact representation schemes for characteristic functions. We first characterize the complexity of CSG under these representation schemes. In this context, the complexity is driven more by the number of rules than by the number of agents. As an initial step toward developing efficient constraint optimization algorithms for solving the CSG problem, we also develop mixed integer programming formulations and show that an off-the-shelf optimization package can perform reasonably well

Heuristic methods for coalition structure generation

The Coalition Structure Generation (CSG) problem requires finding an optimal partition of a set of n agents. An optimal partition means one that maximizes global welfare. Computing an optimal coalition structure is computationally hard especially when there are externalities, i.e., when the worth of a coalition is dependent on the organisation of agents outside the coalition. A number of algorithms were previously proposed to solve the CSG problem but most of these methods were designed for systems without externalities. Very little attention has been paid to finding optimal coalition structures in the presence of externalities, although externalities are a key feature of many real world multiagent systems. Moreover, the existing methods, being non-heuristic, have exponential time complexity which means that they are infeasible for any but systems comprised of a small number of agents. The aim of this research is to develop effective heuristic methods for finding optimal coalition structures in systems with externalities, where time taken to find a solution is more important than the quality of the solution. To this end, four different heuristics methods namely tabu search, simulated annealing, ant colony search and particle swarm optimisation are explored. In particular, neighbourhood operators were devised for the effective exploration of the search space and a compact representation method was formulated for storing details about the multiagent system. Using these, the heuristic methods were devised and their performance was evaluated extensively for a wide range of input data

Sequentially Distributed Coalition Formation Game for Throughput Maximization in C-RANs

Cloud radio access network (C-RAN) has been proposed as a solution to reducing the huge cost of network upgrade while providing the spectral and energy efficiency needed for the new generation cellular networks. In order to reduce the interference that occur in C-RAN and maximize throughput, this paper proposes a sequentially distributed coalition formation (SDCF) game in which players, in this case the remote radio heads (RRHs), can sequentially join multiple coalitions to maximize their throughput. Contrary to overlapping coalition formation (OCF) game where players contribute fractions of their limited resources to different coalitions, the SDCF game offers better stability by allowing sequential coalition formation depending on the availability of resources and therefore providing a balance between efficient spectrum use and interference management. An algorithm for the proposed model is developed based on the merge-only method. The performance of the proposed algorithm in terms of stability, complexity and convergence to final coalition structure is also investigated. Simulation results show that the proposed SDCF game did not only maximize the throughput in the C-RAN, but it also shows better performances and larger capabilities to manage interference with increasing number of RRHs compared to existing methods

Correlation Clustering Based Coalition Formation For Multi-Robot Task Allocation

In this paper, we study the multi-robot task allocation problem where a group of robots needs to be allocated to a set of tasks so that the tasks can be finished optimally. One task may need more than one robot to finish it. Therefore the robots need to form coalitions to complete these tasks. Multi-robot coalition formation for task allocation is a well-known NP-hard problem. To solve this problem, we use a linear-programming based graph partitioning approach along with a region growing strategy which allocates (near) optimal robot coalitions to tasks in a negligible amount of time. Our proposed algorithm is fast (only taking 230 secs. for 100 robots and 10 tasks) and it also finds a near-optimal solution (up to 97.66% of the optimal). We have empirically demonstrated that the proposed approach in this paper always finds a solution which is closer (up to 9.1 times) to the optimal solution than a theoretical worst-case bound proved in an earlier work

Cooperative Games with Bounded Dependency Degree

Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of cooperative games. We show that many computational problems for cooperative games become tractable for games whose dependency degree or supermodular degree are bounded. In particular, we prove that simple games admit efficient algorithms for various solution concepts when the supermodular degree is small; further, we show that computing the Shapley value is always in FPT with respect to the dependency degree. Finally, we note that, while determining the dependency among players is computationally hard, there are efficient algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape