Graph Coalition Structure Generation

Abstract: We give the first analysis of the computational complexity of {\it coalition structure generation over graphs}. Given an undirected graph $G=(N,E)$ and a valuation function $v:2^N\rightarrow\RR$ 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 {\it 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 polynomial time bounds for acyclic, $K_{2,3}$ and $K_4$ minor free graphs. However, as we show, the problem remains NP--complete for planar graphs, and hence, for any $K_k$ minor free graphs where $k\geq 5$. Moreover, our hardness result holds for a particular subclass of valuation functions, termed {\it 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.