Stable partitions in additively separable hedonic games

An important aspect in systems of multiple autonomous agents is the exploitation of synergies via coalition formation. In this paper, we solve various open problems concerning the computational complexity of stable partitions in additively separable hedonic games. First, we propose a polynomial-time algorithm to compute a contractually individually stable partition. This contrasts with previous results such as the NP-hardness of computing individually stable or Nash stable partitions. Secondly, we prove that checking whether the core or the strict core exists is NP-hard in the strong sense even if the preferences of the players are symmetric. Finally, it is shown that verifying whether a partition consisting of the grand coalition is contractually strict core stable or Pareto optimal is coNP-complete.

💡 Research Summary

Additively separable hedonic games model a setting in which each autonomous agent assigns a numeric value to every other agent, and an agent’s utility in a coalition is simply the sum of the values it receives from the members of that coalition. Because agents care only about the composition of their own coalition, the model is attractive for studying coalition formation in multi‑agent systems, but the combinatorial explosion of possible partitions makes the computation of stable outcomes a central challenge.

The paper makes three major contributions. First, it introduces the notion of contractually individually stable (CIS) partitions. A deviation by a single agent is allowed only if every member of the agent’s current coalition consents to the departure. This “contractual consent” captures realistic scenarios where existing members must agree before a member can leave. The authors show that, despite the additional constraint, a CIS partition can be found in polynomial time. Their algorithm proceeds greedily: agents are processed in order of decreasing marginal gain, and each agent moves to the coalition that yields the highest increase in utility provided that the current coalition’s members all agree to the move. Because utilities are additive, each move can be evaluated in O(n) time, and the whole procedure runs in O(n²) time. This result stands in stark contrast to the known NP‑hardness of finding individually stable (IS) or Nash stable (NS) partitions, demonstrating that the contractual requirement actually simplifies the search space.

Second, the paper establishes strong NP‑hardness for the existence of both the core and the strict core, even when the agents’ preferences are symmetric (i.e., v_{ij}=v_{ji}). Using a reduction from the strongly NP‑complete 3‑Partition problem, the authors construct a symmetric additively separable game in which a coalition blocks a given partition if and only if a subset of numbers sums exactly to a target value. Consequently, deciding whether any core‑stable (or strict‑core‑stable) partition exists is as hard as solving 3‑Partition, and the hardness persists regardless of the magnitude of the numeric values (strong NP‑hardness). This result closes an open question about the complexity of core existence under symmetry and shows that even the most permissive core concepts remain computationally intractable.

Third, the paper investigates the status of the grand coalition (the partition consisting of a single coalition containing all agents). It proves that checking whether the grand coalition is contractually strict‑core stable (CSC‑stable) or Pareto optimal is coNP‑complete. The reduction maps unsatisfiable instances of SAT to instances where no blocking coalition exists, thereby showing that a “no‑blocking” certificate is hard to verify. Conversely, a certificate that a blocking coalition exists can be checked in polynomial time, which places the complement problem in NP and yields coNP‑completeness for the verification tasks. Thus, even the most natural and socially desirable outcome—the grand coalition—cannot be efficiently certified as stable or Pareto efficient under these criteria.

Overall, the paper provides a nuanced map of the computational landscape for stability concepts in additively separable hedonic games. By introducing CIS and delivering a concrete polynomial‑time algorithm, it offers a practically useful tool for real‑time coalition restructuring. At the same time, the strong NP‑hardness results for core existence and the coNP‑completeness of grand‑coalition verification underline the inherent difficulty of achieving stronger notions of stability. These findings suggest that future work should focus on approximation algorithms, parameterized complexity analyses, and empirical studies to identify tractable subclasses or heuristic methods that can be deployed in large‑scale multi‑agent environments.