The maximum feasible subset problem (maxFS) is this: given an infeasible set of constraints, find a largest cardinality subset that admits a feasible solution. This problem is NP-hard but has been studied extensively for the case of linear constraints, and good heuristic solution algorithms are available. There is a surprisingly large range of applications for algorithms that solve the linear maxFS problem, including analyzing infeasible linear programs, finding the data depth, placing separating hyperplanes in classification decision trees, recovering sparse data in compressed sensing, dimension reduction in nonnegative matrix factorization, etc. This paper reviews maxFS solution algorithms, and surveys the existing and new applications.

在最大的可行子集的问题（maxFS）是这样的：给定一个不可行集约束，找到一个最大基数子集，它承认一个可行的解决方案。该问题是NP难题，但已针对线性约束进行了广泛研究，并且可以使用良好的启发式求解算法。解决线性maxFS问题的算法的应用范围非常广泛，包括分析不可行的线性程序，查找数据深度，在分类决策树中放置分离的超平面，在压缩感测中恢复稀疏数据，在非负矩阵分解中进行降维，本文回顾了maxFS解决方案算法，并概述了现有和新应用程序。