Sequential testing problems involve a complex system with several components, each of which is "working" with some independent probability. The outcome of each component can be determined by performing a test, which incurs some cost. The overall system status is given by a function $f$ of the outcomes of its components. The goal is to evaluate this function $f$ by performing tests at the minimum expected cost. While there has been extensive prior work on this topic, provable approximation bounds are mainly limited to simple functions like ``k-out-of-n'' and halfspaces. We consider significantly more general "score classification" functions, and provide the first constant factor approximation algorithm (improving over a previous logarithmic approximation ratio). Moreover, our policy is non adaptive: it just involves performing tests in an a priori fixed order. We also consider the related halfspace evaluation problem, where we want to evaluate some function on $d$ halfspaces (e.g., intersection of halfspaces). We show that our approach provides an $O(d^2\log d)$-approximation algorithm for this problem. Our algorithms also extend to the setting of "batched'' tests, where multiple tests can be performed simultaneously while incurring an extra setup cost. Finally, we perform computational experiments that demonstrate the practical performance of our algorithm for score classification. We observe that, for most instances, the cost of our algorithm is within $50\%$ of an information-theoretic lower bound on the optimal value.

翻译：序贯测试问题涉及一个由若干组件构成的复杂系统，每个组件以独立概率处于“正常工作”状态。每个组件的状态可通过执行一次测试来确定，测试需消耗一定成本。系统整体状态由组件结果的函数 $f$ 表示。目标是通过以最小期望成本执行测试来评估该函数 $f$。尽管已有大量关于此课题的研究，但可证明的近似界主要局限于诸如“k-out-of-n”和半空间等简单函数。我们考虑更具一般性的“分数分类”函数，并首次提出常数因子近似算法（改进了此前对数近似比）。此外，我们的策略是非自适应的：仅需按预设的固定顺序执行测试。我们还研究了相关的半空间评估问题，即对 $d$ 个半空间上的某函数（例如半空间交集）进行评估。我们证明了该方法可为该问题提供 $O(d^2\log d)$ 的近似算法。我们的算法还可扩展到“批量测试”场景——即同时执行多个测试但需承担额外设置成本。最后，我们通过计算实验展示了算法在分数分类中的实际性能。实验表明，对于大多数实例，算法成本与最优值的信息论下界偏差不超过50%。