发表机构
Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究刻画了二元行动多智能体合约最优设计在总替代奖励类中的复杂性，证明OXS奖励APX完全、WMRFs有EPTAS、划分拟阵秩函数有FPTAS，并揭示超奖励函数难以近似，强调子模性的关键作用。
AI 中文摘要
我们研究了多智能体二元行动模型中最优合约设计的计算复杂性，重点关注总替代奖励函数及相关类别。虽然加性奖励承认FPTAS，一般子模奖励仅承认常数因子近似，但总替代这一中间类别的复杂性在很大程度上仍未解决。我们揭示了该类别内一个细粒度的近似景观。我们首先证明，即使对于OXS奖励——总替代奖励的一个严格子类，最优合约问题是APX完全的，排除了总替代的PTAS，除非$\mathsf{P}=\mathsf{NP}$。相比之下，对于加权拟阵秩函数（WMRFs）——总替代的另一个自然子类，我们获得了EPTAS，并表明一般情况下不存在随机FPTAS。我们进一步确定了划分（加权）拟阵秩函数的特殊情况，为此我们获得了FPTAS。这与多智能体多行动设置形成对比，在后者中，即使对于未加权的划分拟阵秩函数，也不存在PTAS。最后，我们考虑了更广泛的超奖励函数类别。虽然超奖励在具有单个智能体的相关组合合约模型中保留了总替代的可处理性，但我们在多智能体模型中显示出鲜明对比：没有使用值查询的多项式时间随机算法能够实现期望中的$2^{o(n)}$近似。总之，我们的结果揭示了总替代内部及之外几个性质不同的计算机制，并确定子模性是多智能体合约可近似性的关键成分。
英文摘要
We study the computational complexity of optimal contract design in the multi-agent binary-action model, focusing on gross-substitutes reward functions and related classes. While additive rewards admit an FPTAS and general submodular rewards admit only constant-factor approximation, the complexity within the intermediate class of gross substitutes has remained largely open. We uncover a fine-grained approximation landscape within this class. We first show that the optimal contract problem is APX-complete even for OXS rewards - a strict subclass of gross substitutes rewards, ruling out a PTAS for gross substitutes unless $\mathsf{P}=\mathsf{NP}$. In contrast, for weighted matroid rank functions (WMRFs) - another natural subclass of gross substitutes - we obtain an EPTAS and show that no randomized FPTAS exists in general. We further identify the special case of partition (weighted) matroid rank functions, for which we obtain an FPTAS. This stands in contrast to the multi-agent multi-action setting, where no PTAS exists even for unweighted partition matroid rank functions. Finally, we consider the broader class of ultra reward functions. While ultra rewards retain the tractability of gross substitutes in a related combinatorial contract model with a single agent, we show a sharp contrast in the multi-agent model: no polynomial-time randomized algorithm using value queries can achieve a $2^{o(n)}$-approximation in expectation. Together, our results reveal several qualitatively distinct computational regimes within and beyond gross substitutes, and identify submodularity as a crucial ingredient for the approximability of multi-agent contracts.
