Simple Algorithms for Stochastic Score Classification with Small Approximation Ratios
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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,…,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√(2)≈ 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√(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.
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