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Discrimination-aware classification methods remedy socioeconomic disparities exacerbated by machine learning systems. In this paper, we propose a novel data pre-processing technique that assigns weights to training instances in order to reduce discrimination without changing any of the inputs or labels. While the existing reweighing approach only looks into sensitive attributes, we refine the weights by utilizing both sensitive and insensitive ones. We formulate our weight assignment as a linear programming problem. The weights can be directly used in any classification model into which they are incorporated. We demonstrate three advantages of our approach on synthetic and benchmark datasets. First, discrimination reduction comes at a small cost in accuracy. Second, our method is more scalable than most other pre-processing methods. Third, the trade-off between fairness and accuracy can be explicitly monitored by model users. Code is available athttps://github.com/frnliang/refined_reweighing.