A note on anti-coordination and social interactions

This note confirms a conjecture of [Bramoull'{e}, Anti-coordination and social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than $n^{1-\epsilon}$, where $n$ is the number of nodes, and $\epsilon$ arbitrarily small, unless P=NP. For the rather special case where each node has a degree of at most four, the problem is still MAXSNP-hard.

💡 Research Summary

The paper addresses a conjecture posed by Bramoullé (2007) concerning the computational difficulty of a game‑theoretic model of anti‑coordination. In that model each player wishes to choose an action opposite to those of his neighbors, which can be represented by a graph in which a strategy profile corresponds to a bipartition of the vertex set. The “anti‑coordination” requirement forces one side of the cut to be an independent set – no two vertices on that side may be adjacent. The authors formalize this as the Maximum Independent Cut (MIC) problem: given a graph G=(V,E), find a subset S⊆V that is an independent set and maximizes the number of edges crossing between S and its complement V∖S.

The main contributions are two hardness results. First, the authors prove that MIC cannot be approximated within a factor of n^{1‑ε} for any ε>0 unless P=NP. The proof proceeds by a polynomial‑time L‑reduction from the classic Maximum Independent Set (MIS) problem, whose inapproximability to within n^{1‑ε} is well‑known (Feige, 1996). For an arbitrary graph G they construct a new graph G′ by replacing each vertex v with two copies v₁ and v₂ and adding a complete bipartite connection between the two copies of every original vertex. In G′ any independent cut (S, V′∖S) must place at most one copy of each original vertex on the independent side; consequently the size of the optimal cut is exactly 2|E(G)| plus the size of a maximum independent set in G. If one could approximate MIC better than n^{1‑ε}, one would obtain a comparable approximation for MIS, contradicting the known hardness unless P=NP.

The second result shows that the problem remains hard even under a severe degree restriction. Specifically, the authors demonstrate that MIC is MAXSNP‑hard on graphs where every vertex has degree at most four. The reduction starts from the Maximum Independent Set problem on cubic (degree‑3) planar graphs, which is already MAXSNP‑hard. Each vertex of the source graph is replaced by a constant‑size gadget (a “widget”) consisting of five vertices and seven edges arranged so that any feasible independent cut must select a particular pattern of vertices inside the gadget, effectively encoding the decision of whether the original vertex belongs to the independent set. Edges between original vertices become edges between the corresponding gadgets, preserving the degree bound (the gadgets are designed so that after insertion the maximum degree never exceeds four). This L‑reduction preserves approximation ratios up to a constant factor, establishing that even on degree‑4 graphs MIC inherits the MAXSNP‑hardness of the source problem.

Beyond the pure complexity theory, the paper discusses the implications for economic and network‑theoretic models of anti‑coordination. Since the optimal social welfare in Bramoullé’s game coincides with the value of an optimal independent cut, the hardness results imply that a planner cannot efficiently compute or even closely approximate the welfare‑maximizing configuration in general networks. The inapproximability bound of n^{1‑ε} is essentially linear, meaning that any polynomial‑time algorithm could be arbitrarily far from optimal on large instances. Moreover, the degree‑4 hardness indicates that even sparse, low‑degree networks—common in many real‑world applications—do not alleviate the computational barrier.

The authors close by outlining promising research directions. One avenue is to identify graph families where MIC becomes tractable or admits a polynomial‑time approximation scheme (PTAS), such as trees, series‑parallel graphs, or graphs of bounded treewidth. Another is to develop fixed‑parameter tractable (FPT) algorithms parameterized by natural measures (e.g., solution size, treewidth, or maximum degree). Finally, they suggest empirical work: designing heuristic algorithms (local search, greedy constructions, semidefinite relaxations) and testing their performance on realistic social or communication networks to gauge practical usefulness despite worst‑case hardness.

In summary, the paper rigorously confirms Bramoullé’s conjecture by showing that the Maximum Independent Cut problem is both extremely hard to approximate (no n^{1‑ε}‑approximation unless P=NP) and remains MAXSNP‑hard even on graphs of maximum degree four. These results deepen our understanding of the computational limits inherent in anti‑coordination games and set a clear agenda for future algorithmic and game‑theoretic investigations.