We study here several variants of the covariates fine balance problem where we generalize some of these problems and introduce a number of others. We present here a comprehensive complexity study of the covariates problems providing polynomial time algorithms, or a proof of NP-hardness. The polynomial time algorithms described are mostly combinatorial and rely on network flow techniques. In addition we present several fixed-parameter tractable results for problems where the number of covariates and the number of levels of each covariate are seen as a parameter.

中文翻译：

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协变量精细平衡变体的算法和复杂性

我们在这里研究了协变量精细平衡问题的几种变体，其中我们概括了其中一些问题并介绍了一些其他问题。我们在这里对协变量问题进行了全面的复杂性研究，提供多项式时间算法，或 NP 硬度的证明。所描述的多项式时间算法大多是组合的，并且依赖于网络流技术。此外，对于协变量的数量和每个协变量的水平数被视为参数的问题，我们提出了几个固定参数的易处理结果。