Abstract
Deploying multiple large language models (LLMs) in parallel to classify an unknown ground-truth label is a common practice, yet the problem of optimally allocating queries across heterogeneous models remains poorly understood. In this paper, we formulate a robust, offline query-planning problem that minimizes total query cost subject to statewise error constraints which guarantee reliability for every possible ground-truth label. We first establish that this problem is NP-hard via a reduction from the minimum-weight set cover problem. To overcome this intractability, we develop a surrogate by combining a union bound decomposition of the multi-class error into pairwise comparisons with Chernoff-type concentration bounds. The resulting surrogate admits a closed-form, multiplicatively separable expression in the query counts and is guaranteed to be feasibility-preserving. We further show that the surrogate is asymptotically tight at the optimization level: the ratio of surrogate-optimal cost to true optimal cost converges to one as error tolerances shrink, with an explicit rate of O\left(\log\log(1/\alpha_{\min}) / \log(1/\alpha_{\min})\right). Finally, we design an asymptotic fully polynomial-time approximation scheme (AFPTAS) that returns a surrogate-feasible query plan within a (1+\varepsilon) factor of the surrogate optimum.
