Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer (2010). An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, they might offer multiple levels of a service, or they may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings where each agent's services are either fully acquired or not at all. The main reason for this, is that in settings with partial allocations like the ones mentioned, there are strong inapproximability results. Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results in this work. We design a polynomial-time, deterministic, truthful, budget-feasible $(2+\sqrt{3})$-approximation mechanism for the setting where each agent offers multiple levels of service and the auctioneer has a discrete separable concave valuation function. We then use this result to design a deterministic, truthful and budget-feasible mechanism for the setting where any fraction of a service can be acquired and the auctioneer's valuation function is separable concave (i.e., the sum of concave functions). The approximation ratio of this mechanism depends on how `nice' the concave functions are, and is $O(1)$ for valuation functions that are sums of $O(1)$-regular functions (e.g., functions like $\log(1+x)$). For the special case of a linear valuation function, we improve the best known approximation ratio for the problem from $1+\phi$ (by Klumper & Sch\"afer (2022)) to $2$. This establishes a separation between this setting and its indivisible counterpart.

翻译：预算可行采购自Singer（2010）提出以来，已成为机制设计领域的重要范式。具有严格预算约束的拍卖方（买方）希望向一组策略性智能体（卖方）采购商品或服务。在许多场景中，允许拍卖方仅部分购买智能体所提供的服务具有实际意义——例如，某智能体可能拥有多份待售商品、提供多级服务，或可在指定时间区间内承担任意比例的任务。然而，相关文献主要聚焦于智能体服务要么全额收购、要么完全不收购的场景。其根本原因在于，上述具有部分分配特征的场景存在强不可近似性结论。在可负担每个智能体全部服务的温和假设下，本研究成功规避了此类局限。我们针对每个智能体提供多级服务、拍卖方持有离散可分凹估值函数的场景，设计了一个多项式时间、确定性、真实且预算可行的（2+√3）近似机制。随后利用该成果，为可获取任意比例服务且拍卖方估值函数为可分凹函数（即多个凹函数之和）的场景，设计了一个确定性、真实且预算可行的机制。该机制的近似比取决于凹函数的性质：对于由O(1)正则函数（如log(1+x)）求和构成的估值函数，其近似比为O(1)。针对线性估值函数的特例，我们将该问题的最佳已知近似比从1+φ（Klumper & Schäfer, 2022）改进至2，从而揭示了该场景与不可分场景之间的本质差异。