Coalition structure generation over graphs

  • Authors:

  • Thomas Voice;Maria Polukarov;Nicholas R. Jennings

  • Affiliations:

  • School of Electronics and Computer Science, University of Southampton, UK;School of Electronics and Computer Science, University of Southampton, UK;School of Electronics and Computer Science, University of Southampton, UK and Department of Computing and Information Technology, King Abdulaziz University, Saudi Arabia

  • Venue:
  • Journal of Artificial Intelligence Research
  • Year:
  • 2012

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Abstract

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 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.