We study the general setting of weighted search in which a number of weighted targets are hidden in a star-like environment, and a mobile searcher must locate a subset of targets with aggregate weight at least a given value W. The cost of the strategy is the distance traversed by the searcher, and its performance is measured by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy. This is the first study of a setting that generalizes several problems in search theory such as searching for a single target and searching for unit-weighted targets. We present and analyze a near-optimal strategy using an approach based on parameterized analysis. This problem formulates settings of resource allocation among parallel tasks under uncertainty; specifically, we demonstrate further applications in the design of interruptible systems based on adaptive scheduling of contract algorithms.