This paper studies how a small number of expensive, reliable “oracle” correctors can steer an unreliable multi-agent swarm toward a truthful consensus. Modeling consensus over a graph, it defines coherence as H(R)=tr M(R)^−1, where oracle placement and strength affect the system at a cost. The authors claim that the objective remains submodular even with heterogeneous oracle strengths, enabling a cost-benefit greedy strategy with a 1−1/e approximation. They derive budget-correctness results for complete graphs and report task-dependent concave or convex cost-quality behavior across Qwen3 models from 0.6B to 32B parameters.
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