Abstract. 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 invokations 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 SUMe and MAXe 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 SUMe and MAXe objectives on continuous and MAXe 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 SUMe and MAXe objectives facilitates simultaneous approximation with respect to both. Since one-time and continuous testing can be viable alternatives, we study the overhead of continuous testing, captured by the ratio of optimal one-time to continuous cover times. We establish that the worst-case ratio is O(log n), but also provide evidence, by considering Zipf distributions, that the typical ratio is lower.