Individual Rationality in Constrained Hedonic Games: Friends, Enemies, and Neutrals

Abstract: We study constrained coalition formation in games induced by friends, enemies, and neutrals, under the two standard refinements of additively separable preferences: friend-oriented and enemy-oriented. We ask for partitions that are individually rational (IR), while additionally requiring exactly $k$ non-empty coalitions, each satisfying a prescribed lower and upper bound on its size. Although IR alone is trivial to satisfy for any hedonic game, the size constraints make it computationally intractable to decide whether a feasible partition exists. The two models tell strikingly different stories. Under enemy-oriented preferences, the problem collapses to size-constrained graph coloring, and its complexity follows accordingly. Under friend-oriented preferences, however, the picture is far more intricate, and is governed by the enmity structure rather than the friendships. The complexity is further shaped by two factors: how strict the imposed size requirements are, and whether relationships are symmetric or asymmetric, with several cases turning out tractable in the symmetric setting but intractable once asymmetry is allowed. Charting this boundary in terms of both classical and parameterized complexity, we provide a complete understanding of which properties of the friend/enemy structure are responsible for hardness.

• The paper develops a near-complete complexity classification of individually rational partitions with fixed coalition counts and size bounds, covering four constraint variants under friend- and enemy-oriented preferences.
• Enemy-oriented instances reduce to size-constrained graph coloring, while friend-oriented cases range from polynomial-time solvable for small coalitions or bounded structure to NP-complete under modest coalition sizes and few enemies.
• The paper provides treewidth-, vertex-cover-, and clique-width-based algorithms and hardness results, showing that symmetry can enable tractability while asymmetry, recluses, and size constraints create major computational barriers.

The problem and its motivation

This paper studies coalition formation in hedonic games where each agent classifies every other agent as a friend, an enemy, or a neutral. The authors ask whether there exists a partition of the agents into coalitions that is individually rational (IR)—no agent prefers being alone to their assigned coalition—while simultaneously satisfying structural constraints on the outcome: exactly $k$ non-empty coalitions, each with prescribed lower and upper bounds on size. Although IR alone is trivially satisfiable by the all-singletons partition, the addition of size constraints makes the decision problem non-trivial, and the paper's central contribution is a near-complete complexity classification of when feasible IR partitions exist.

The model builds on the friends–enemies–neutrals framework of Dimitrov et al. and Ohta et al., with two canonical refinements of additively separable preferences: friend-oriented preferences (an agent first maximizes friends in their coalition, then minimizes enemies) and enemy-oriented preferences (an agent first minimizes enemies, then maximizes friends). Under these restrictions, IR admits simple combinatorial characterizations: under enemy-oriented preferences, a structure is IR iff no agent shares a coalition with an enemy; under friend-oriented preferences, iff each agent either has a friend or no enemy in their coalition. Four constraint variants are studied in increasing generality: the $k$-coalition variant ($k$-HG), balanced-size coalitions (BSC), fixed-size coalitions (FSC), and fully size-constrained coalitions (SCC).

Enemy-oriented preferences collapse to coloring

Under enemy-oriented preferences the analysis resolves completely: friendships play no role, and all four variants reduce to size-constrained variants of graph $k$-coloring on the enmity graph. Consequently, IR-EO-$k$ and IR-EO-BSC are NP-complete even for $k=3$ with every agent having exactly four enemies, while IR-EO-SCC for $k=2$ is solvable in linear time via dynamic programming over connected components of the enmity graph. The practical implication is that any algorithmic advance on constrained coloring transfers directly to this setting, and vice versa.

Friend-oriented preferences: hardness from few coalitions and small sizes

The friend-oriented case is substantially richer. A strict dichotomy emerges in terms of the number of desired coalitions: if $k=1$, IR-FO-SCC is decidable in linear time (one checks whether the grand coalition is IR), but IR-FO-$k$ and IR-FO-BSC are NP-complete already for $k=2$, even with symmetric preferences, at most eight friends per agent, and only two friendless agents. The reduction from Monotone 4-Regular NAE 3-SAT uses two guard agents who must occupy different coalitions, forcing clause agents into coalitions where they need at least one friend—encoding truth assignments through coalition membership.

Coalition size also yields a sharp threshold. If every coalition has upper bound at most 2, IR-FO-SCC reduces to finding a matching of appropriate size in the "non-enmity" graph and is polynomial-time solvable. In contrast, IR-FO-BSC is NP-complete even when every coalition has size exactly 3, symmetric preferences hold, each agent has three or four friends, and no neutrals exist; the reduction is from $k$0-Decomposition on subcubic graphs, with agents representing edges and friendship encoding edge adjacency.

Recluses, symmetry, and asymmetry

A recurring theme is that agents without friends—the paper calls them recluses—are the principal source of intractability, since they require enemy-free coalitions. For IR-FO-$k$1 with symmetric preferences and no recluses, a polynomial-time algorithm exists: it partitions along connected components of the friendship graph and peels leaves off spanning trees to reach exactly $k$2 non-empty coalitions while preserving IR. Strikingly, this tractability evaporates under asymmetry: IR-FO-$k$3 remains NP-complete even without recluses once directed friendships are allowed, via a reduction from 3-Coloring using long directed friendship cycles that force dummy agents into singletons. Symmetry thus acts as a genuine computational resource. The symmetric case with exactly one recluse is left open.

Structural parameters of the enmity graph

Restricting the enmity graph's vertex cover number produces a fine-grained picture. If $k$4, IR-FO-$k$5 is solvable in linear time, and for symmetric instances even IR-FO-SCC is polynomial-time solvable via careful case analysis around the single covering agent. However, IR-FO-BSC is NP-complete with $k$6, $k$7, at most one enemy and two friends per agent in the asymmetric setting—a strong negative result showing that a single universally disliked agent plus balanced-size requirements suffices for hardness; the reduction encodes Independent Set on regular graphs through friendship cycles whose inclusion patterns correspond to independent sets. On the positive side, for symmetric instances IR-FO-$k$8 is polynomial-time solvable up to $k$9. Hardness resumes at $k$0 for symmetric instances, and at $k$1 for asymmetric ones; IR-FO-BSC is NP-hard on symmetric instances already at $k$2. Additionally, IR-FO-FSC is fixed-parameter tractable parameterized by the number of conflicting agents (those with at least one enemy), since deleting friendships among non-conflicting agents leaves a relationship graph with vertex cover equal to that number.

Treewidth, clique-width, and tight lower bounds

For the full relationship graph $k$3, the paper gives an FPT algorithm for IR-FO-$k$4 parameterized by treewidth, running in time $k$5 via dynamic programming over a nice tree decomposition with signatures tracking bag-to-coalition assignments, per-agent flags encoding whether an agent has a friend or enemy in their partial coalition, and coalition non-emptiness. This running time is optimal unless ETH fails, established by a reduction from $k$6-Coloring preserving treewidth. Extending the DP to track coalition sizes yields an XP algorithm for IR-FO-SCC parameterized by $k$7—but not FPT, since IR-FO-BSC is W[1]-hard parameterized by $k$8 combined, and W[1]-hard parameterized by clique-width for both IR-FO-$k$9 and IR-FO-BSC. Finally, exploiting the stronger vertex cover number of $k$0, the authors design an FPT algorithm for IR-FO-FSC parameterized by $k$1, combining exhaustive guessing of the cover's partition with a maximum-flow computation that allocates remaining agents subject to lower bounds.

Limitations and open questions

Several cases remain unresolved, most notably whether IR-FO-SCC is FPT parameterized by the vertex cover number of the relationship graph, and the status of the symmetric one-recluse case for IR-FO-$k$2. The positive results for SCC variants are weaker than for other variants (XP rather than FPT under treewidth-based parameters), reflecting genuine W[1]-hardness barriers rather than technical gaps. The paper also notes that its results extend prior work of Fioravantes et al. on ASHGs, but that the earlier algorithms based on $k$3-fold integer programming do not transfer here, since expressing friend-oriented valuations additively requires weights linear in the number of agents.

Conclusion

The paper delivers a comprehensive classical and parameterized complexity classification of individually rational coalition formation under size constraints for friends–enemies–neutrals preferences. Its two central findings are that under friend-oriented preferences it is the enmity structure—not the friendships—that governs complexity, and that symmetry of relationships is a decisive computational resource whose removal flips several tractable cases to NP-hardness. The remaining open cases, particularly the parameterized complexity of the general size-constrained variant, delineate the precise boundary of current knowledge in this area.