We address auctions in two-sided markets with budget constraints on buyers, a fundamental setting also crucial for applications such as display advertising. Our goal is to design efficient mechanisms that satisfy dominant strategy incentive compatibility, individual rationality, and budget balance. To overcome the limitations of impossibility theorems, we assume prior knowledge of sellers' valuations and focus on liquid welfare, an efficiency objective that takes budgets into account. Our contributions are twofold: First, we improve the efficiency guarantees of the polyhedral clinching auction by Hirai and Sato (2022). Second, using the reduction method of D"{u}tting et al. (2021), we extend the mechanism to an efficient single-sample mechanism for budget-constrained auctions, providing the budget extension of their results. Notably, our results hold even under polymatroid constraints and apply to both divisible and indivisible goods.

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