This paper presents an algorithmic study of a class of covering mixed-integer linear programming problems which encompasses classic cover problems, including multidimensional knapsack, facility location and supplier selection problems. We first show some properties of optimal solutions, which are then used to decompose the problem into instances of the multidimensional knapsack cover problem with a single continuous variable per dimension. The proposed decomposition is used to design a polynomial-time approximation scheme for the problem with a fixed number of constraints. To the best of our knowledge, this is the first approximation scheme for such a general class of covering mixed-integer linear programs. Moreover, we design a fully polynomial-time approximation scheme and an approximate linear programming formulation for the case with a single constraint. These results improve upon the previously best-known 2-approximation algorithm for the knapsack cover problem with a single continuous variable. Finally, we show a perfect compact formulation for the case where all variables have the same lower and upper bounds. Analogous results are derived for the packing and more general variants of the problem.

翻译：本文对一类覆盖混合整数线性规划问题进行了算法研究，该类问题涵盖了经典覆盖问题，包括多维背包、设施选址和供应商选择问题。我们首先展示了最优解的一些性质，并利用这些性质将问题分解为每维度仅含一个连续变量的多维背包覆盖问题实例。所提出的分解方法被用于为具有固定约束数量的问题设计多项式时间近似方案。据我们所知，这是针对如此一般类别的覆盖混合整数线性规划问题的首个近似方案。此外，针对单约束情形，我们设计了完全多项式时间近似方案和近似线性规划公式。这些结果改进了先前针对单连续变量背包覆盖问题的最佳已知2-近似算法。最后，我们展示了当所有变量具有相同上下界时的完美紧凑公式。对于问题的包装及更一般变体，我们也推导出了类似结果。