The main contribution of this paper resides in providing novel algorithmic advances and analytical insights for the sequential hiring problem, a recently introduced dynamic optimization model where a firm adaptively fills a limited number of positions from a pool of applicants with known values and acceptance probabilities. While earlier research established a strong foundation -- notably an LP-based $(1 - \frac{e^{-k}k^k}{k!})$-approximation by Epstein and Ma (Operations Research, 2024) -- the attainability of superior approximation guarantees has remained a central open question. Our work addresses this challenge by establishing the first polynomial-time approximation scheme for sequential hiring, proposing an $O(n^{O(1)} \cdot T^{2^{\tilde{O}(1/ε^{2})}})$-time construction of semi-adaptive policies whose expected reward is within factor $1 - ε$ of optimal. To overcome the constant-factor optimality loss inherent to earlier literature, and to circumvent intrinsic representational barriers of adaptive policies, our approach is driven by the following innovations: -- The block-responsive paradigm: We introduce block-responsive policies, a new class of decision-making strategies, selecting ordered sets (blocks) of applicants rather than single individuals, while still allowing for internal reactivity. -- Adaptivity and efficiency: We prove that these policies can nearly match the performance of general adaptive policies while utilizing polynomially-sized decision trees. -- Efficient construction: By developing a recursive enumeration-based framework, we resolve the problematic ``few-positions'' regime, bypassing a fundamental hurdle that hindered previous approaches.

翻译：本文的主要贡献在于为顺序招聘问题提供了新颖的算法进展与分析见解。该问题是一个近期提出的动态优化模型，其中一家公司从已知价值与接受概率的申请人池中，自适应地填补有限数量的职位。尽管早期研究奠定了坚实基础——特别是 Epstein 和 Ma（《运筹学》，2024）提出的基于线性规划的 $(1 - \frac{e^{-k}k^k}{k!})$-近似——但能否获得更优的近似保证一直是一个核心开放性问题。我们的工作通过为顺序招聘问题建立首个多项式时间近似方案来应对这一挑战，提出了一个 $O(n^{O(1)} \cdot T^{2^{\tilde{O}(1/ε^{2})}})$ 时间构造的半自适应策略，其期望收益在最优解的 $1 - ε$ 因子范围内。为了克服早期文献固有的常数因子最优性损失，并规避自适应策略的内在表示障碍，我们的方法由以下创新驱动：——块响应范式：我们引入了块响应策略，这是一种新的决策策略类别，选择有序的申请人集合（块）而非单个个体，同时仍允许内部反应性。——自适应性及效率：我们证明这些策略在利用多项式规模决策树的同时，几乎能匹配一般自适应策略的性能。——高效构造：通过开发基于递归枚举的框架，我们解决了棘手的“少数职位”情形，绕过了阻碍先前方法的一个根本性障碍。