Graph Coalition Structure Generation

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.

💡 Research Summary

The paper introduces and rigorously studies the Graph Coalition Structure Generation (GCSG) problem, a natural extension of the classic coalition structure generation (CSG) problem to settings where the agents (or resources) are embedded in an undirected graph. In GCSG, a coalition must induce a connected subgraph of the underlying graph G = (N,E), and the goal is to partition the vertex set N into such connected coalitions so that the sum of their values is maximized.

A central modeling assumption is the Independent of Disconnected Members (IDM) property of the valuation function v : 2^N → ℝ. IDM requires that for any two vertices i and j that are separated by a vertex separator C, the marginal contribution of i to C does not depend on whether j is present. This captures the intuition that agents that are not linked by a communication or trust path cannot affect each other’s contribution. A particularly important subclass of IDM functions is the edge‑sum valuation, where each edge (i,j) carries a weight w_{ij} and the value of a coalition C is simply Σ_{(i,j)∈E, i,j∈C} w_{ij}.

The authors first establish that GCSG is NP‑complete on general graphs, even when the valuation is of the edge‑sum type. The hardness proof reduces from 3‑SAT by constructing a planar graph whose vertices correspond to literals and a distinguished vertex s. Positive‑weight edges connect s to every literal node, while large negative‑weight edges enforce consistency (no coalition may contain both a literal and its negation). The optimal coalition structure attains value m (the number of clauses) if and only if the original formula is satisfiable. Consequently, the problem remains NP‑hard on planar graphs, and therefore on any K_k‑minor‑free class with k ≥ 5.

On the algorithmic side, the paper provides a baseline exhaustive algorithm for arbitrary graphs that runs in O(n²·e + nⁿ) time and O(n²) space. The algorithm enumerates all subgraphs with at most n − 1 edges (i.e., all possible spanning forests), computes their connected components, and evaluates the total coalition value. Although exponential, this establishes an upper bound that matches the problem’s inherent difficulty.

The core technical contribution is a separator‑based divide‑and‑conquer framework (Theorem 1). For a graph class S that is closed under taking subgraphs and admits an f(n)‑separator theorem with constant α < 1, the authors show that if the number of possible connected coalition structures on any n‑vertex graph from S is bounded by a function g(n), then an optimal GCSG solution can be found in O(exp(h(β,n))) time for any β > α, where

h(β,n) = ⌊log n / |log β|⌋ ·