Coalition Structure Generation over Graphs

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) \to 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 K_k minor free graphs where k \geq 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m^2) 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.

💡 Research Summary

The paper investigates the computational complexity of the Coalition Structure Generation (CSG) problem when the agents are embedded in an undirected graph and each coalition must induce a connected subgraph. Formally, given a graph G = (N, E) and a valuation function v : 2^N → ℝ, the task is to partition the node set N into disjoint, connected subsets (coalitions) so that the sum of the coalition values Σ_{C∈𝒫} v(C) is maximized. The authors first establish that this graph‑constrained CSG problem remains NP‑complete in the general case, extending the known hardness of the unconstrained version.

A central contribution is the identification of a broad class of valuation functions called “Independence of Disconnected Members” (IDM). An IDM function guarantees that the marginal contribution of a node does not depend on nodes that are not in the same connected component. Even under this restriction, the authors prove NP‑hardness by a reduction from 3‑SAT. They construct, for any 3‑SAT instance with m clauses, a planar graph with O(m²) vertices and an associated CSG instance such that the SAT formula is satisfiable if and only if the CSG instance attains a value above a prescribed threshold. This construction preserves planarity, thereby showing that the problem is NP‑complete for planar graphs and consequently for any K_k‑minor‑free class with k ≥ 5.

The paper further narrows the focus to the “edge‑sum” subclass of IDM functions, where the value of a coalition is simply the sum of the weights of the edges in the induced subgraph: v(S) = Σ_{(u,v)∈E