# Coalition Structure Generation over Graphs

arXiv: https://arxiv.org/abs/1401.6428 · HTML: https://mlsanity.com/paper/1401.6428/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2014.

## Built on by (most cited first, 5 of 5)

- [GCS-Q: Quantum Graph Coalition Structure Generation](https://mlsanity.com/paper/2212.11372/) (2022)
- [A 4-approximation algorithm for min max correlation clustering](https://mlsanity.com/paper/2310.09196/) (2023)
- [A Polyhedral Study of Lifted Multicuts](https://mlsanity.com/paper/2202.08068/) (2022)
- [An attention model for the formation of collectives in real-world domains](https://mlsanity.com/paper/2205.00215/) (2022)
- [A Graph Multi-separator Problem for Image Segmentation](https://mlsanity.com/paper/2307.04592/) (2023)

## Abstract (arXiv)

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.