Optimal Coalition Structures in Graph Games

We consider the problem of finding the optimal coalition structure in \emph{Weighted Graph Games (WGGs)}, a simple restricted representation of coalitional games [Deng, Papadimitriou, Math. OR 1994]. The agents in such games are vertices of the graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The \emph{optimal coalition structure} is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minor-free and bounded degree graphs.

[1]  K. Wagner.  Über eine Eigenschaft der ebenen Komplexe , 1937 .

[2]  Nicholas R. Jennings,et al.  Overlapping Coalition Formation for Efficient Data Fusion in Multi-Sensor Networks , 2006, AAAI.

[3]  R. J. Aumann,et al.  Cooperative games with coalition structures , 1974 .

[4]  Mamata Samal,et al.  Approximation algorithm for correlation clustering , 2009, 2009 First International Conference on Networked Digital Technologies.

[5]  Victor R. Lesser,et al.  Coalitions Among Computationally Bounded Agents , 1997, Artif. Intell..

[6]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[7]  Yoav Shoham,et al.  Multi-attribute coalitional games , 2006, EC '06.

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  J. M. Bilbao,et al.  Cooperative Games on Combinatorial Structures , 2000 .

[10]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[11]  Haris Aziz,et al.  Complexity of coalition structure generation , 2011, AAMAS.

[12]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[13]  Sarvapali D. Ramchurn,et al.  Anytime Optimal Coalition Structure Generation , 2007, AAAI.

[14]  Sarit Kraus,et al.  Methods for Task Allocation via Agent Coalition Formation , 1998, Artif. Intell..

[15]  Avrim Blum,et al.  Correlation Clustering , 2004, Machine Learning.

[16]  Onn Shehory,et al.  Coalition structure generation with worst case guarantees , 2022 .

[17]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[18]  Ken-ichi Kawarabayashi,et al.  Algorithmic graph minor theory: Decomposition, approximation, and coloring , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[19]  Andrew Thomason,et al.  The Extremal Function for Complete Minors , 2001, J. Comb. Theory B.

[20]  Jinsong Tan,et al.  A note on the inapproximability of correlation clustering , 2007, Inf. Process. Lett..

[21]  D. Yun Yeh,et al.  A Dynamic Programming Approach to the Complete Set Partitioning Problem , 1986, BIT.

[22]  Erik D. Demaine,et al.  Diameter and Treewidth in Minor-Closed Graph Families, Revisited , 2004, Algorithmica.

[23]  Kyomin Jung,et al.  Coalitional Structure Generation in Skill Games , 2010, AAAI.

[24]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[25]  Jeffrey S. Rosenschein,et al.  Coalitional skill games , 2008, AAMAS.

[26]  B. Peleg,et al.  Introduction to the Theory of Cooperative Games , 1983 .

[27]  Nicholas R. Jennings,et al.  An improved dynamic programming algorithm for coalition structure generation , 2008, AAMAS.

[28]  Vincent Conitzer,et al.  Coalition Structure Generation Utilizing Compact Characteristic Function Representations , 2009, CP.

[29]  Nicholas R. Jennings,et al.  Coalition Structures in Weighted Voting Games , 2008, ECAI.

[30]  Vincent Conitzer,et al.  Coalition Structure Generation Utilizing Compact Characteristic Function Representations , 2011 .

[31]  Martin Szummer,et al.  A Graphical Model for Simultaneous Partitioning and Labeling , 2005, AISTATS.