Coalition Structure Generation Problem Research Articles

Constrained optimal grouping of cloud application components

Cloud applications are built from a set of components often deployed as containers, which can be deployed individually on separate Virtual Machines (VMs) or grouped on a smaller set of VMs. Additionally, the application owner may have inhibition constraints regarding the co-location of components. Finding the best way to deploy an application means finding the best groups of components and the best VMs, and it is not trivial because of the complexity coming from the number of possible options. The problem can be mapped onto may known combinatorial problems as binpacking and knapsack formulations. However, these approaches often assume homogeneus resources and fail to incorporate the inhibition constraints. The main contribution of this paper are firstly a novel formulation of the grouping problem as constrained Coalition Structure Generation (CSG) problem, including the specification of the value function which fulfills the criteria of a Characteristic Function Game (CFG). The CSG problem aims to determine stable and disjoint groups of players collaborating to optimize the joint outcome of the game, and a CFG is a common representation of a CSG, where each group is assigned a value and where the value of the game is the sum of the groups’ contributions. Secondly, the Integer-Partition (IP) CSG algorithm has been modified and extended to handle constraints. The proposed approach is evaluated with the extended IP algorithm, and a novel exhaustive search algorithm establishing the optimum grouping for comparison. The evaluation shows that our approach with the modified algorithm evaluates on average significantly less combinations than the CSG state-of-the-art algorithm. The proposed approach is promising for optimized constrained Cloud application management as the modified IP algorithm can optimally solve constrained grouping problems of attainable sizes.

On the Online Coalition Structure Generation Problem

We consider the online version of the coalition structure generation problem, in which agents, corresponding to the vertices of a graph, appear in an online fashion and have to be partitioned into coalitions by an authority (i.e., an online algorithm). When an agent appears, the algorithm has to decide whether to put the agent into an existing coalition or to create a new one containing, at this moment, only her. The decision is irrevocable. The objective is partitioning agents into coalitions so as to maximize the resulting social welfare that is the sum of all coalition values. We consider two cases for the value of a coalition: (1) the sum of the weights of its edges, and (2) the sum of the weights of its edges divided by its size.
 Coalition structures appear in a variety of application in AI, multi-agent systems, networks, as well as in social networks, data analysis, computational biology, game theory, and scheduling. For each of the coalition value functions we consider the bounded and unbounded cases depending on whether or not the size of a coalition can exceed a given value α. Furthermore, we consider the case of a limited number of coalitions and various weight functions for the edges, i.e., unrestricted, positive and constant weights. We show tight or nearly tight bounds for the competitive ratio in each case.

An Improved Algorithm for Optimal Coalition Structure Generation

The Coalition Structure Generation (CSG) problem is a partitioning of a set of agents into exhaustive and disjoint coalitions to maximize social welfare. This NP-complete problem arises in many practical scenarios. Prominent examples are included in the field of transportation, e-Commerce, distributed sensor networks, and others. The fastest exact algorithm to solve the CSG problem is ODP-IP, which is a hybrid version of two previously established algorithms, namely Improved Dynamic Programming (IDP) and IP. In this paper, we show that the ODP-IP algorithm performs many redundant operations. To improve ODP-IP, we propose a faster abortion mechanism to speed up IP’s search. Our abortion mechanism decides at runtime which of the IP's operations are redundant to skip them. Then, we propose a modified version of IDP (named MIDP) and an improved version of IP (named IIP). Based on these two improved algorithms, we develop a hybrid version (MIDP-IIP) to solve the CSG problem. After a detailed description of the new algorithm MIDP-IIP, an experimental comparison is conducted against ODP-IP. Our analysis shows that MIDP-IIP performs fewer operations than ODP-IP. In addition, MIDP-IIP reduced significantly many problem instances running times (11% to 37 %), and improved drastically some of them.

FACS: Fast Code-based Algorithm for Coalition Structure Generation (Student Abstract)

In this paper, we propose a new algorithm for the Coalition Structure Generation (CSG) problem that can be run with more than 28 agents while using a complete set of coalitions as input. The current state-of-the-art limit for exact algorithms to solve the CSG problem within a reasonable time is 27 agents. Our algorithm uses a novel representation of the search space and a new code-based search technique. We propose an effective heuristic search method to efficiently explore the space of coalition structures using our code based technique and show that our method outperforms existing state-of-the-art algorithms by multiple orders of magnitude.

BOSS: A Bi-directional Search Technique for Optimal Coalition Structure Generation with Minimal Overlapping (Student Abstract)

In this paper, we focus on the Coalition Structure Generation (CSG) problem, which involves finding exhaustive and disjoint partitions of agents such that the efficiency of the entire system is optimized. We propose an efficient hybrid algorithm for optimal coalition structure generation called BOSS. When compared to the state-of-the-art, BOSS is shown to perform better by up to 33.63% on benchmark inputs. The maximum time gain by BOSS is 3392 seconds for 27 agents.

Improving coalition structure search with an imperfect algorithm: analysis and evaluation results

Optimal Coalition Structure Generation (CSG) is a significant research problem in multi-agent systems that remains difficult to solve. This problem has many important applications in transportation, eCommerce, distributed sensor networks and others. The CSG problem is NP-complete and finding the optimal result for n agents needs to check $$O (n^n)$$ possible partitions. The ODP–IP algorithm (Michalak et al. in Artif Intell 230:14–50, 2016) achieves the current lowest worst-case time complexity of $$O (3^n)$$ . In the light of its high computational time complexity, we devise an Imperfect Dynamic Programming (ImDP) algorithm for the CSG problem with runtime $$O (n2^n)$$ given n agents. Imperfect algorithm means that there are some contrived inputs for which the algorithm fails to give the optimal result. We benchmarked ImDP against ODP–IP and proved its efficiency. Experimental results confirmed that ImDP algorithm performance is better for several data distributions, and for some it improves dramatically ODP–IP. For example, given 27 agents, with ImDP for agent-based uniform distribution time gain is 91% (i.e. 49 min).

A New Genetic Algorithm Encoding for Coalition Structure Generation Problems

Genetic algorithms have proved to be a useful improvement heuristic for tackling several combinatorial problems, including the coalition structure generation problem. In this case, the focus lies on selecting the best partition from a discrete set. A relevant issue when designing a Genetic algorithm for coalition structure generation problems is to choose a proper genetic encoding that enables an efficient computational implementation. In this paper, we present a novel hybrid encoding, and we compare its performance against several genetic encoding proposed in the literature. We show that even in difficult instances of the coalition structure generation problem, the proposed approach is a competitive alternative to obtaining good quality solutions in reasonable computing times. Furthermore, we also show that the encoding relevance increases as the number of players increases.

An anytime algorithm for optimal simultaneous coalition structure generation and assignment

An important research problem in artificial intelligence is how to organize multiple agents, and coordinate them, so that they can work together to solve problems. Coordinating agents in a multi-agent system can significantly affect the system’s performance—the agents can, in many instances, be organized so that they can solve tasks more efficiently, and consequently benefit collectively and individually. Central to this endeavor is coalition formation—the process by which heterogeneous agents organize and form disjoint groups (coalitions). Coalition formation often involves finding a coalition structure (an exhaustive set of disjoint coalitions) that maximizes the system’s potential performance (e.g., social welfare) through coalition structure generation. However, coalition structure generation typically has no notion of goals. In cooperative settings, where coordination of multiple coalitions is important, this may generate suboptimal teams for achieving and accomplishing the tasks and goals at hand. With this in mind, we consider simultaneously generating coalitions of agents and assigning the coalitions to independent alternatives (e.g., tasks/goals), and present an anytime algorithm for the simultaneous coalition structure generation and assignment problem. This combinatorial optimization problem has many real-world applications, including forming goal-oriented teams. To evaluate the presented algorithm’s performance, we present five methods for synthetic problem set generation, and benchmark the algorithm against the industry-grade solver CPLEX using randomized data sets of varying distribution and complexity. To test its anytime-performance, we compare the quality of its interim solutions against those generated by a greedy algorithm and pure random search. Finally, we also apply the algorithm to solve the problem of assigning agents to regions in a major commercial strategy game, and show that it can be used in game-playing to coordinate smaller sets of agents in real-time.

A Task-Oriented Heuristic for Repairing Infeasible Solutions to Overlapping Coalition Structure Generation

Overlapping coalition formation (OCF), which provides a natural framework for modeling scenarios where each agent can join and allocate their resources to several completely different coalitions at the same time, has become a very active topic in multiagent systems. For OCF in resource-constrained and subadditive task oriented domains, an agent may not possess sufficient resources to meet the needs of multiple coalitions simultaneously. As a result, there may exist many potential resource conflicts among the rival overlapping coalitions. To tackle such situations, we first present a natural variation of the traditional OCF model and analyze the size of the solution space and the computational complexity of the overlapping coalition structure generation (OCSG) problem. Next, we develop a generic task-oriented heuristic (TOH) for individual repairs that can be used in binary meta-heuristic algorithms to generate overlapping coalitions in a parallel manner. Moreover, we show how the proposed TOH repairs a 2-D individual to resolve resource conflicts and discuss several basic properties. Finally, to evaluate the effectiveness of TOH, we compare it with the existing agent-oriented heuristic for the OCSG problem. The empirical results demonstrate that TOH is of high efficiency and effectiveness in harsh environments with fierce competition over scarce resources.

Improved WPM encoding for coalition structure generation under MC-nets

The Coalition Structure Generation (CSG) problem plays an important role in the domain of coalition games. Its goal is to create coalitions of agents so that the global welfare is maximized. To date, Weighted Partial MaxSAT (WPM) encoding has shown high efficiency in solving the CSG problem, which encodes a set of constraints into Boolean propositional logic and employs an off-the-shelf WPM solver to find out the optimal solution. However, in existing WPM encodings, a number of redundant encodings are asserted. This results in additional calculations and correspondingly incurs performance penalty. Against this background, this paper presents an Improved Rule Relation-based WPM (I-RWPM) encoding for the CSG problem, which is expressed by a set of weighted rules in a concise representation scheme called Marginal Contribution net (MC-net). In order to effectively reduce the constraints imposed on encodings, we first identify a subset of rules in an MC-net, referred as a set of freelance rules. We prove that solving the problem made up of all freelance rules can be achieved with a straightforward means without any extra encodings. Thus the set of rules requiring to be encoded is downsized. Next, we improve the encoding of transitive relations among rules. To be specific, compared with the existing rule relation-based encoding that generates transitive relations universally among all rules, I-RWPM only considers the transitivity among rules with particular relationship. In this way, the number of constraints to be encoded can be further decreased. Experiments suggest that I-RWPM significantly outperforms other WPM encodings for solving the same set of problem instances.

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.

Coalition Structure Generation Utilizing Graphical Representation of Partition Function Games

Forming effective coalition is a central research challenge in AI and multi-agent systems. The Coalition Structure Generation (CSG) problem is well-known as one of major research topics in coalitional games. The CSG problem is to partition a set of agents into coalitions so that the sum of utilities is maximized. This paper studies a CSG problem for partition function games (PFGs), where the value of a coalition differs depending on the formation of other coalitions generated by non-member agents. Traditionally, in PFGs, the input of a coalitional game is a black-box function called a partition function that maps an embedded coalition (a coalition and the coalition structure) to its value. Recently, a novel concise representation scheme called the Partition Decision Trees (PDTs) has been proposed. The PDTs is a graphical representation based on multiple rules. In this paper, we propose new algorithms that can solve a CSG problem by utilizing PDTs representation. More specifically, we modify PDTs representation to effectively handle negative value rules and apply the depth-first branch and bound algorithm. We experimentally show that our algorithm can solve a CSG problem well.

A hierarchical clustering approach to large-scale near-optimal coalition formation with quality guarantees

Coalition formation is a fundamental approach to multi-agent coordination, and a key challenge in this context is the coalition structure generation problem, where a set of agents must be partitioned into the best set of coalitions. This problem is NP-hard and typical optimal algorithms do not scale to more than 50 agents: efficient approximate solutions are therefore needed for hundreds or thousands of agents. In this paper we propose a novel heuristic, based on ideas and tools used in the data clustering domain. In particular, we present a coalition formation algorithm inspired by the well known class of hierarchical agglomerative clustering techniques (Linkage algorithms). We present different variants of the algorithm, which we call Coalition Linkage (C-Link) and demonstrate how such algorithm can be adapted to graph restricted coalition formation problems (where an interaction graph defined among the agents restricts the set of feasible coalitions). Moreover, we discuss how we can provide an upper bound on the value of the optimal coalition structure, and we show that for specific characteristic functions we can provide such bounds while maintaining polynomial computational costs and memory requirements. We empirically evaluate the different variants of the C-Link algorithm on two synthetic benchmark data-sets, as well as in two real world scenarios, involving a collective energy purchasing and a ride-sharing application. In these settings C-Link achieves promising results providing high quality solutions and solving problem involving thousands of agents in few minutes.

Using Computational Intelligence Algorithms to Solve the Coalition Structure Generation Problem in Coalitional Skill Games

Coalitional skill games (CSGs) are a simple model of cooperation in an uncertain environment where each agent has a set of skills that are required to accomplish a variety of tasks and each task requires a set of skills to be completed, but each skill is very hard to be quantified and can only be qualitatively expressed. Thus far, many computational questions surrounding CSGs have been studied. However, to the best of our knowledge, the coalition structure generation problem (CSGP), as a central issue of CSGs, is extremely challenging and has not been well solved. To this end, two different computational intelligence algorithms are herein evaluated: binary particle swarm optimization (BPSO) and binary differential evolution (BDE). In particular, we develop the two stochastic search algorithms with two-dimensional binary encoding and corresponding heuristic for individual repairs. After that, we discuss some fundamental properties of the proposed heuristic. Finally, we compare the improved BPSO and BDE with the state-of-the-art algorithms for solving CSGP in CSGs. The experimental results show that our algorithms can find the same near optimal solutions with the existing approaches but take extremely short time, especially under the large problem size.

Coalition structure generation problems: optimization and parallelization of the IDP algorithm in multicore systems

SummaryThe coalition structure generation problem is well known in the area of multi‐agent systems. Its goal is to establish coalitions between agents while maximizing the global welfare. Among the existing different algorithms designed to solve the coalition structure generation problem, DP and IDP are the ones with smaller temporal complexity. After analyzing the operation of the dynamic programming and improved dynamic programming algorithms, we have identified which are the most frequent operations and propose an optimized method. In addition, we study and implement a method for dividing the work into different threads. To describe incremental improvements of the algorithm design, we first compare performance of an improved single central processing unit core version where we obtain speedups ranging from 7 × to 11 × . Then, we describe the best resource use in a multi‐thread optimized version where we obtain an additional 7.5 × speedup running in a 12‐core machine.

A hybrid exact algorithm for complete set partitioning

In the Complete Set Partitioning problem we are given a finite set of elements where every subset is associated with a value, and the goal is to partition this set into disjoint subsets so as to maximise the sum of subset values. This abstract problem captures the Coalition Structure Generation problem in cooperative games in characteristic function form, where each subset, or coalition, of agents can make a profit when working together, and the goal is to partition the set of agents into coalitions to maximise the total profit. It also captures the special case of the Winner Determination problem in combinatorial auctions, where bidders place bids on every possible bundle of goods, and the goal is to find an allocation of goods to bidders that maximises the profit of the auctioneer.The main contribution of this article is an extensive theoretical analysis of the search space of the Complete Set Partitioning problem, which reveals that two fundamentally different exact algorithms can be significantly improved upon in terms of actual runtime. These are (1) a dynamic programming algorithm called “DP” [48,36] and (2) a tree-search algorithm called “IP” [32]. We start by drawing a link between DP and a certain graph describing the structure of the search space. This link reveals that many of DP's operations are in fact redundant. Consequently, we develop ODP—an optimal version of DP that avoids all of its redundant operations. Since ODP and IP are based on different design paradigms, each has its own strengths and weaknesses compared to the other. Thus, one has to trade off the advantages of one algorithm for the advantages of the other. This raises the following question: Is this trade-off inevitable? To answer this question, we develop a new representation of the search space, which links both algorithms, and allows for contrasting the workings of the two. This reveals that ODP and IP can actually be combined, leading to the development of ODP-IP—a hybrid algorithm that avoids the limitations of its constituent parts, while retaining and significantly improving upon the advantages of each part.We benchmark our algorithm against that of Björklund et al. (2009) [8], which runs in O(2n) time given n agents. We observe that the algorithm of Björklund et al. relies on performing arithmetic operations with very large integers, and assumes that any such operation has unit cost. In practice, however, working with large integers on a modern PC is costly. Consequently, when implemented, our O(3n) algorithm outperforms that of Björklund et al. by several orders of magnitude on every problem instance, making ours the fastest exact algorithm for complete set partitioning to date in practice.

A fast approximation algorithm for solving the complete set packing problem

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.

Extending MaxSAT to Solve the Coalition Structure Generation Problem with Externalities Based on Agent Relations

Extending MaxSAT to Solve the Coalition Structure Generation Problem with Externalities Based on Agent Relations

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 other’s marginal con- tribution 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 K_k 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(m^2) 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.

Anytime coalition structure generation in multi-agent systems with positive or negative externalities

Much of the literature on multi-agent coalition formation has focused on Characteristic Function Games, where the effectiveness of a coalition is not affected by how the other agents are arranged in the system. In contrast, very little attention has been given to the more general class of Partition Function Games, where the emphasis is on how the formation of one coalition could influence the performance of other co-existing coalitions in the system. However, these inter-coalitional dependencies, called externalities from coalition formation, play a crucial role in many real-world multi-agent applications where agents have either conflicting or overlapping goals.Against this background, this paper is the first computational study of coalitional games with externalities in the multi-agent system context. We focus on the Coalition Structure Generation (CSG) problem which involves finding an exhaustive and disjoint division of the agents into coalitions such that the performance of the entire system is optimized. While this problem is already very challenging in the absence of externalities, due to the exponential size of the search space, taking externalities into consideration makes it even more challenging as the size of the input, given n agents, grows from O(2n) to O(nn).Our main contribution is the development of the first CSG algorithm for coalitional games with either positive or negative externalities. Specifically, we prove that it is possible to compute upper and lower bounds on the values of any set of disjoint coalitions. Building upon this, we prove that in order to establish a worst-case guarantee on solution quality it is necessary to search a certain set of coalition structures (which we define). We also show how to progressively improve this guarantee with further search.Since there are no previous CSG algorithms for games with externalities, we benchmark our algorithm against other state-of-the-art approaches in games where no externalities are present. Surprisingly, we find that, as far as worst-case guarantees are concerned, our algorithm outperforms the others by orders of magnitude. For instance, to reach a bound of 3 given 24 agents, the number of coalition structures that need to be searched by our algorithm is only 0.0007% of that needed by Sandholm et al. (1999) [1], and 0.5% of that needed by Dang and Jennings (2004) [2]. This is despite the fact that the other algorithms take advantage of the special properties of games with no externalities, while ours does not.