We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly \({\mathcal {N}}{\mathcal {P}}\)-hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

我们研究了精确分离最大深度的覆盖不等式的问题。我们为它的解决方案提出了一个伪多项式时间动态规划算法，由于它，我们证明了这个问题是弱的\（{\ mathcal {N}} {\ mathcal {P}} \）-难（类似于分离最大违规的掩盖不平等的问题）。我们在有和没有冲突约束的情况下，对背负实例和多维背负问题进行了广泛的计算实验。结果表明，使用基于最大深度准则的切割平面生成方法，我们可以通过生成比采用标准最大违反准则时小的切口数量来优化覆盖不等式闭合。我们还介绍了点到超平面距离背包问题（PHD-KP），该问题与最大深度覆盖不等式的分离问题密切相关，并展示了如何将提出的动态规划算法应用于有效解决PHD- KP也是如此。