We study the optimal contract problem in the \emph{combinatorial actions} framework of Dütting et al.~[FOCS'21], where a principal delegates a project to an agent who chooses a subset of hidden, costly actions, and the resulting reward is given by a monotone set function over the actions. The principal offers a contract that specifies the fraction of the reward the agent receives, and the goal is to compute a contract that maximizes the principal's expected utility. Prior work established polynomial-time algorithms for \emph{gross substitutes} rewards, while showing NP-hardness for general submodular rewards; subsequent work extended tractability to \emph{supermodular} rewards, demonstrating that tractable cases exist in both the substitutes and complements regimes. This left open the precise boundary of tractability for the optimal contract problem. Our main result is a polynomial-time algorithm for the optimal contract problem under \Ultra\ rewards, a class that strictly contains gross substitutes but is not confined to subadditive rewards, thereby bridging the substitutes and complements regimes. We further extend our results beyond additive costs, establishing a polynomial-time algorithm for \Ultra\ rewards and cost functions that are the sum of additive and symmetric functions. To the best of our knowledge, this is the first application of \Ultra\ functions in a prominent economic setting.

翻译：我们研究了Dütting等人[FOCS'21]提出的\emph{组合行动}框架下的最优契约问题。在该框架中，委托方将项目委托给代理方，代理方从一组隐藏且具有成本的行为中选择一个子集，最终收益由这些行为上的单调集合函数给出。委托方提供一份规定代理方获得收益份额的契约，目标是计算使委托方期望效用最大化的契约。先前的研究针对\emph{总替代}收益建立了多项式时间算法，同时证明了一般次模收益的NP难性；后续研究将可处理性扩展至\emph{超模}收益，表明在替代与互补机制中均存在可处理情形。这使最优契约问题的精确可处理边界成为开放问题。我们的主要成果是针对\Ultra\收益下的最优契约问题提出多项式时间算法，该收益类别严格包含总替代收益且不限于次可加收益，从而在替代与互补机制间架起桥梁。我们进一步将结果扩展到非加性成本场景，为\Ultra\收益及"加性函数与对称函数之和"构成的成本函数建立了多项式时间算法。据我们所知，这是\Ultra\函数在重要经济场景中的首次应用。