We study linear contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a linear contract with several agents, each of whom can decide to take a costly action or not. The principal observes only the outcome of the agents' collective actions, not the actions themselves, and obtains a reward from this outcome. Agents that take an action incur a cost, and so naturally agents require a fraction of the principal's reward as an incentive for taking their action. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. Our focus is on the case when the agents are vertices in a graph and the principal's reward corresponds to the number of edges between agents who take their costly action. This case represents the natural scenario when an action of each agent complements actions of other agents though collaborations. Recently, Deo-Campo Vuong et.al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless $\mathcal{P} = \mathcal{NP}$. On a positive note, the authors provided an additive PTAS for the case when all agents have the same cost. They asked whether an additive PTAS can be obtained for the general case, i.e when agents potentially have different costs. We answer this open question in positive.

翻译：本研究探讨多智能体环境下组合优化问题的线性契约设计。在该问题中，委托方需设计面向多个智能体的线性契约，每个智能体可自主决定是否执行需付出代价的行动。委托方仅能观测智能体集体行动产生的结果（而非具体行动），并从中获得收益。执行行动的智能体需承担相应成本，因此需要从委托方收益中获取一定比例作为行动激励。委托方需确定分配给各智能体的收益比例，以实现自身期望效用最大化。本文重点研究智能体构成图结构顶点、委托方收益对应于执行代价行动的智能体间边数量的情形，该情形反映了智能体行动通过协作产生互补效应的自然场景。近期Deo-Campo Vuong等人证明，除非$\mathcal{P} = \mathcal{NP}$，否则该问题无法获得任何有限倍数近似解或加性FPTAS。积极方面，作者在全体智能体成本相同时给出了加性PTAS，并针对智能体成本可能相异的一般情形，提出是否存在加性PTAS的开放性问题。本文对该问题给出了肯定性解答。