, Volume 24, Issue 1, pp 25–55 | Cite as

Improved WPM encoding for coalition structure generation under MC-nets

  • Xiaojuan LiaoEmail author
  • Miyuki Koshimura
  • Kazuki Nomoto
  • Suguru Ueda
  • Yuko Sakurai
  • Makoto Yokoo


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.


Weighted partial MaxSAT Coalition structure generation Coalition games 



This work is partially supported by the National Natural Science Foundation of China (Grant Number 61806171), Ministry of Education in China Project of Humanities and Social Sciences (Grant Number 17YJCZH260), JSPS KAKENHI (Grant Numbers JP16K00304, JP17K00307, JP17H00761 and JP18H03299).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chengdu University of TechnologyChengduChina
  2. 2.Kyushu UniversityFukuokaJapan
  3. 3.Saga UniversitySagaJapan
  4. 4.National Institute of Advanced Industrial Science and TechnologyTokyoJapan

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