  Large Language Models (LLMs) have achieved state-of-the-art performance in
text re-ranking. This process includes queries and candidate passages in the
prompts, utilizing pointwise, listwise, and pairwise prompting strategies. A
limitation of these ranking strategies with LLMs is their cost: the process can
become expensive due to API charges, which are based on the number of input and
output tokens. We study how to maximize the re-ranking performance given a
budget, by navigating the vast search spaces of prompt choices, LLM APIs, and
budget splits. We propose a suite of budget-constrained methods to perform text
re-ranking using a set of LLM APIs. Our most efficient method, called EcoRank,
is a two-layered pipeline that jointly optimizes decisions regarding budget
allocation across prompt strategies and LLM APIs. Our experimental results on
four popular QA and passage reranking datasets show that EcoRank outperforms
other budget-aware supervised and unsupervised baselines.
