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Gkenosis 等人(ESA 2018)推出的沙沙分分级问题(SSC)(ESA 2018):我们得到的是美元测试。每个测试美元可以以美元计价,每个测试美元可以独立地以美元/美元进行。此外,我们知道(Integer)间距($0,\dots,n ⁇ 美元到美元)的分割,目的是进行测试,以确定从成功测试数量所在的分区间隔到尽可能降低预期成本的间隔之间的间隔。Ghuge 等人(IPCO 2022)最近显示,存在一个混合时常数正数缩贴近算算算法。我们显示,在两种战略中,分别用美元/j/p_美元和美元(1-p_j美元)的间隔进行测试。正如Gkensisciform等人(Gk 等人)已经建议的那样,在特定情况下,近似比率从6美元到3+2美元之间的差差差比值略下降。我们还显示,在本次测试中,正值的平比值分析中, 3-rqxxxxxxx的平比值越近比值越近比值越近比值越高的平平平平平平平平平平平调。