Heuristic methods for optimal coalition structure generation

The problem of finding the optimal coalition structure arises frequently in multiagent systems. Heuristic approaches for solving this problem are needed because of its computational complexity. This paper studies two such approaches: tabu search and simulated annealing. Through simulations we show that tabu search generates better quality solutions than simulated annealing for coalition games in characteristic function form and those in partition function form

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

Computing optimal coalition structures in polynomial time

The optimal coalition structure determination problem is in general computationally hard. In this article, we identify some problem instances for which the space of possible coalition structures has a certain form and constructively prove that the problem is polynomial time solvable. Specifically, we consider games with an ordering over the players and introduce a distance metric for measuring the distance between any two structures. In terms of this metric, we define the property of monotonicity, meaning that coalition structures closer to the optimal, as measured by the metric, have higher value than those further away. Similarly, quasi-monotonicity means that part of the space of coalition structures is monotonic, while part of it is non-monotonic. (Quasi)-monotonicity is a property that can be satisfied by coalition games in characteristic function form and also those in partition function form. For a setting with a monotonic value function and a known player ordering, we prove that the optimal coalition structure determination problem is polynomial time solvable and devise such an algorithm using a greedy approach. We extend this algorithm to quasi-monotonic value functions and demonstrate how its time complexity improves from exponential to polynomial as the degree of monotonicity of the value function increases. We go further and consider a setting in which the value function is monotonic and an ordering over the players is known to exist but ordering itself is unknown. For this setting too, we prove that the coalition structure determination problem is polynomial time solvable and devise such an algorithm

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

Optimal coalition structure generation on large-scale renewable energy smart grids

Most renewable energy sources are dependent on unpredictable weather conditions, which have considerable variation over space and time. The intermittent nature of this production means that any renewable energy prosumer may sometimes produce an amount of energy in excess of its local consumption needs and sometimes in deficiency. This thesis is concerned with developing methods that can improve the effectiveness and widespread adoption of renewable energy usage. In order for renewable energy to be more economically viable, there needs to be a scheme for sharing energy among the prosumers so that those with excess energy can give their excess amounts to those in energy deficiency. That is the task addressed in this thesis. The way to deal with this problem is to setup an optimal arrangement of local coalitions of renewable energy prosumers such that energy is shared within the coalitions in an optimally efficient manner. As is formally explained early on in this work, finding such an optimal coalition arrangement is an example of a Coalition Structure Generation (CSG) problem. The most straightforward way to find an optimal solution for a given pool of prosumer agents in these circumstances is to examine every possible coalition partition (coalition structure) and evaluate its comparative utility. This is known as ``exhaustive search'' (ES) and can be computationally expensive. As has been shown earlier, the number of such evaluations in ES even for a pool of twenty agents can be in the tens of trillions. The problem for us in the renewable energy domain is that, because of the constantly changing weather conditions among the scattered prosumers, the CSG optimization calculation must be carried out every hour of the day. This means that the ES approach in the CSG optimization calculation for a reasonable number of prosumer agents is computationally intractable. So a more computationally feasible stochastic optimization method must be used, which searches through the coalition structure search space in order to find a reasonably good solution even if it is not the global optimum. To this end, a number of stochastic optimization search methods have been investigated in this thesis, including some of our own novel extensions to existing approaches. These search methods have been examined with respect to two different connection arrangements with respect to the outside world – (1) when the local prosumer networks have a connection to a public utility power grid and can therefore buy needed energy (at a high price) from the grid and sell excess energy (at a low price) to the grid and (2) when the local prosumer networks are isolated from any public utility, which is referred to as ``island mode''. The overall goal of these investigations has been to find an optimization approach that arrives at a near-optimal (near the global optimum of the given search space) that is computationally efficient (i.e. it does not require a vast amount of computer memory or running time). Based on these empirical examinations, which have employed realistic parameters drawn from existing consumption and renewable energy data sets, the following conclusions concerning renewable energy can be drawn from this study: • It is feasible to employ ordinary computer resources to obtain on an hourly basis near-optimal energy-sharing coalition structures that will lead to the more effective and economical use of renewable energy. • This energy-sharing approach will contribute to more rapid adoption and proliferation of existing renewable energy equipment and infrastructure. The principal contributions towards these end that this thesis work has made are as follows: • A modelling framework has been set up that can be used for extensive empirical determinations of near-optimal energy-sharing coalition structures. • A detailed empirical study has been carried out that has examined the relative capabilities in this context of various optimal coalition structure search methods, including genetic algorithms (GA), dynamic programming (DP), particle swarm optimization (PSO), population-based incremental learning (PBIL), and several variants to PBIL. • The novel extensions to basic PBIL optimization have included Top-k Merit Weighting PBIL (PBIL-MW), Set-ID Encoding Schemes, and Hierarchical PBIL-MW

Coalition Structure Generation with GRASP

The coalition structure generation problem represents an active research area in multi-agent systems. A coalition structure is defined as a partition of the agents involved in a system into disjoint coalitions. The problem of finding the optimal coalition structure is NP-complete. In order to find the optimal solution in a combinatorial optimization problem it is theoretically possible to enumerate the solutions and evaluate each. But this approach is infeasible since the number of solutions often grows exponentially with the size of the problem. In this paper we present a greedy adaptive search procedure (GRASP) to efficiently search the space of coalition structures in order to find an optimal one