We revisit the Stochastic Score Classification (SSC) problem introduced by Gkenosis et al. (ESA 2018): We are given $n$ tests. Each test $j$ can be conducted at cost $c_j$, and it succeeds independently with probability $p_j$. Further, a partition of the (integer) interval $\{0,\dots,n\}$ into $B$ smaller intervals is known. The goal is to conduct tests so as to determine that interval from the partition in which the number of successful tests lies while minimizing the expected cost. Ghuge et al. (IPCO 2022) recently showed that a polynomial-time constant-factor approximation algorithm exists. We show that interweaving the two strategies that order tests increasingly by their $c_j/p_j$ and $c_j/(1-p_j)$ ratios, respectively, -- as already proposed by Gkensosis et al. for a special case -- yields a small approximation ratio. We also show that the approximation ratio can be slightly decreased from $6$ to $3+2\sqrt{2}\approx 5.828$ by adding in a third strategy that simply orders tests increasingly by their costs. The similar analyses for both algorithms are nontrivial but arguably clean. Finally, we complement the implied upper bound of $3+2\sqrt{2}$ on the adaptivity gap with a lower bound of $3/2$. Since the lower-bound instance is a so-called unit-cost $k$-of-$n$ instance, we settle the adaptivity gap in this case.

翻译：我们重新审视由Gkenosis等人（ESA 2018）提出的随机分数分类（SSC）问题：给定$n$个测试，每个测试$j$可以以成本$c_j$执行，并以概率$p_j$独立成功。此外，已知整数区间$\{0,\dots,n\}$被划分为$B$个更小子区间。目标是执行测试，以确定成功测试数量落在划分中的哪个子区间，同时最小化期望成本。Ghuge等人（IPCO 2022）近期证明存在一个多项式时间的常数因子近似算法。我们证明，将两种分别按$c_j/p_j$和$c_j/(1-p_j)$比率递增排列测试的策略交叉结合——正如Gkenosis等人在特殊情况下所提出的——能获得较小的近似比。我们还证明，通过添加第三种按成本递增排列测试的策略，近似比可从$6$略微降至$3+2\sqrt{2}\approx 5.828$。两种算法的分析虽不平凡，但可称为简洁。最后，我们将$3+2\sqrt{2}$的适应度间隙上界与$3/2$的下界互补。由于下界实例是所谓的单位成本$k$-of-$n$实例，我们在此情形下确定了适应度间隙。