Approximating Combinatorial Contracts with Arbitrary Costs

Abstract

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.