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We address the optimal sink scheduling problem in wireless sensor networks (WSNs). The problem is inherently difficult since sink scheduling and data routing are tightly coupled. Previous approaches either have questionable performance due to no joint considerations, or are based on relaxed constraints. Our aim is to fill in this blank in the research. First, by discretizing continuous time, we develop a novel bound technique to connect time-varying routes with the placement of sinks. This bounding technique transforms time-related constraints into pattern-based ones and allows us to mathematically formulate this optimization in a pattern-based way. The complexity of directly solving this optimization is intractable; therefore, on the basis of column generation (CG), a computationally efficient algorithm is developed to reduce the complexity by decomposing the problem into sub-problems and iteratively solving them to approach optimality. Simulations demonstrate the efficiency of the algorithm and substantiate the importance of sink mobility in energy-constrained sensor networks.