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.

中文翻译：

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改进的WPM编码，用于在MC-net下生成联盟结构

联盟结构生成（CSG）问题在联盟博弈领域中起着重要作用。其目标是建立代理商联盟，以使全球福利最大化。迄今为止，加权部分MaxSAT（WPM）编码在解决CSG问题方面显示出了很高的效率，该算法将一组约束编码为布尔命题逻辑，并使用现成的WPM求解器来找出最佳解决方案。但是，在现有的WPM编码中，会声明许多冗余编码。这导致额外的计算并相应地导致性能损失。在这种背景下，本文提出了一种针对CSG问题的改进的基于规则关系的WPM（I-RWPM）编码，该编码由一组加权规则以简洁的表示方式表示，称为边际贡献网（MC-net）。为了有效减少对编码的限制，我们首先在MC-net中确定规则的子集，称为自由职业规则集。我们证明，可以用一种简单的方法来解决由所有自由职业规则组成的问题，而无需任何额外的编码。因此，需要编码的规则集被缩小。接下来，我们改进规则之间的传递关系的编码。具体而言，与现有的基于规则关系的编码在所有规则之间普遍生成传递关系相比，I-RWPM仅考虑具有特定关系的规则之间的传递性。这样，可以进一步减少要编码的约束的数量。实验表明，在解决同一组问题实例方面，I-RWPM明显优于其他WPM编码。