Scheduling subset tests: One-time, continuous, and how they relate

A test scheduling instance is specified by a set of elements, a set of tests, which are subsets of elements, and numeric priorities assigned to elements. The schedule is a sequence of test invocations with the goal of covering all elements. This formulation had been used to model problems in multiple application domains from network failure detection to broadcast scheduling. The modeling considered both SUM_e and MAX_e objectives, which correspond to average or worst-case cover times over elements (weighted by priority), and both one-time testing, where the goal is to detect if a fault is currently present, and continuous testing, performed in the background in order to detect presence of failures soon after they occur. Since all variants are NP hard, the focus is on approximations. We present combinatorial approximations algorithms for both SUM_e and MAX _e objectives on continuous and MAX_e on one-time schedules. The approximation ratios we obtain depend logarithmically on the number of elements and significantly improve over previous results. Moreover, our unified treatment of SUM_e and MAX_e objectives facilitates simultaneous approximation with respect to both. Since one-time and continuous testing can be viable alternatives, we study their relation, which captures the overhead of continuous testing. We establish that for both SUM_e and MAX_e objectives, the ratio of the optimal one-time to continuous cover times is O(log n), where n is the number of elements. We show that this is tight as there are instances with ratio Ω(log n). We provide evidence, however, by considering Zipf distributions, that the typical ratio is lower.