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 (see, e.g., Chan & Chen (2014), Anari et al. (2018)). 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 $2$.

翻译：预算可行的采购自Singer(2010)引入以来，已成为机制设计中的主要范式。具有严格预算约束的拍卖者（买方）有意向从一组策略性代理（卖方）群体中购买商品或服务。在许多场景中，允许拍卖者仅部分购买代理提供的服务是合理的，例如代理可能拥有多个待售商品副本、提供多级服务，或在特定时间段内可执行任务的任意部分。然而，相关文献主要聚焦于每个代理的服务要么被完全采购要么完全不采购的场景。其主要原因在于，在存在部分分配（如上述情形）的场景中，存在强不可近似性结果（参见Chan & Chen(2014)、Anari等(2018)）。在能够完全支付每个代理的温和假设下，本研究规避了此类结果。我们针对每个代理提供多级服务且拍卖者具有离散可分离凹估值函数的场景，设计了多项式时间、确定性、真实、预算可行的$(2+\sqrt{3})$-近似机制。进而利用该结果，为任意比例的服务可被采购且拍卖者估值函数为可分离凹函数（即凹函数之和）的场景，设计了确定性、真实且预算可行的机制。该机制的近似比取决于凹函数的"优良"程度，对于由$O(1)$-正则函数（例如$\log(1+x)$类函数）之和构成的估值函数，近似比为$O(1)$。在线性估值函数的特殊情况下，我们将该问题已知最优近似比从$1+\phi$（由Klumper & Schäfer(2022)提出）改进至$2$。