Simple Algorithms for Stochastic Score Classification with Small Approximation Ratios
Benedikt M. Plank, Kevin Schewior · SIAM Journal on Discrete Mathematics · 2024
Abstract. We revisit the Stochastic Score Classification (SSC) problem introduced by Gkenosis et al. (ESA 2018): We are given [Formula: see text] tests. Each test [Formula: see text] can be conducted at cost [Formula: see text], and it succeeds independently with probability [Formula: see text]. Further, a partition of the (integer) interval [Formula: see text] into [Formula: see text] 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, Gupta, and Nagarajan (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 [Formula: see text] and [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] on the adaptivity gap with a lower bound of 3/2. Since the lower-bound instance is a so-called unit-cost [Formula: see text]-of-[Formula: see text] instance, we settle the adaptivity gap in this case.