近似具有任意成本的组合合约
Approximating Combinatorial Contracts with Arbitrary Costs
- City University of Hong Kong（香港城市大学）
- Peking University（北京大学）
- Chinese Academy of Sciences（中国科学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对线性支付下的单智能体组合合约，提出一种与成本无关的确定性近似算法，通过收益侧尺度证书和几何搜索，以 $O(n\log(n+1)/\varepsilon)$ 次供给查询实现 $(1-\varepsilon)$ 近似，去除了对成本函数的结构假设。
AI中文摘要：
我们研究线性支付下的单智能体组合合约。在收益份额 $\alpha\in[0,1]$ 下，智能体从 $n$ 个隐藏动作中选择一个子集 $S$，产生收益 $f(S)$ 并承担成本 $c(S)$，以最大化 $\alpha f(S)-c(S)$，而委托人获得 $(1-\alpha)f(S)$。对于非负加性收益和单调超模成本，Dütting 等人（SODA 2026）证明了精确优化的指数级供给查询下界，并留下了一个开放问题：是否存在表示无关的近似算法。我们以更强的形式解决了这个问题，去除了对成本函数的所有结构假设：对于非负加性收益、任意归一化非负集合成本，以及每个 $\varepsilon\in(0,1)$，我们给出一个确定性 $(1-\varepsilon)$-近似算法，使用 $O(n\log(n+1)/\varepsilon)$ 次供给查询，查询复杂度与数值位长和断点分离度无关。更一般地，在精确最佳响应和值查询访问下，相同的保证和渐近查询复杂度适用于归一化单调次加性收益和任意归一化非负成本。关键思想是一个与成本无关、基于收益侧的尺度证书。算法尝试所有单例收益作为候选锚点；其中一个将最优保留份额括在 $n^2$ 因子内。几何搜索和诱导响应收益的单调性随后产生近似，而无需枚举最佳响应断点。
英文摘要：
We study single-agent combinatorial contracts under linear payments. Under a reward share $α\in[0,1]$, an agent chooses a subset $S$ of $n$ hidden actions, generating reward $f(S)$ at cost $c(S)$, to maximize $αf(S)-c(S)$, while the principal receives $(1-α)f(S)$. For nonnegative additive rewards and monotone supermodular costs, Dütting et al. (SODA 2026) proved an exponential supply-query lower bound for exact optimization and left open whether a representation-independent approximation is possible. We resolve this question in a stronger form, removing all structural assumptions on the cost function: for nonnegative additive rewards, arbitrary normalized nonnegative set costs, and every $\varepsilon\in(0,1)$, we give a deterministic $(1-\varepsilon)$-approximation using $O(n\log(n+1)/\varepsilon)$ supply queries, with query complexity independent of numerical bit lengths and breakpoint separation. More generally, the same guarantee and asymptotic query complexity hold for normalized monotone subadditive rewards and arbitrary normalized nonnegative costs under exact best-response and value-query access. The key idea is a cost-independent, reward-side scale certificate. The algorithm tries all singleton rewards as candidate anchors; one of them brackets the optimal retained share within a factor $n^2$. Geometric search and monotonicity of the induced response reward then yield the approximation without enumerating best-response breakpoints.
