On the exact separation of cover inequalities of maximum-depth

Fabio Furini,Stefano Coniglio,Daniele Catanzaro

doi:10.1007/s11590-021-01741-0

Abstract

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.

Highlights

The separation of maximum-violation inequalities is the standard approach for generating valid inequalities in Mixed Integer Linear Programs (MILPs) [33]
We 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
We have proposed a pseudopolynomial-time Dynamic Programming (DP) algorithm for the maximumdepth separation of such inequalities, which has allowed us to established that the problem is weakly N P-hard

Summary

Introduction

The separation of maximum-violation inequalities is the standard approach for generating valid inequalities in Mixed Integer Linear Programs (MILPs) [33]. 5) is the characterization of the computational complexity of this problem To this end, we develop an exact pseudopolynomial-time dynamic programming algorithm by means of which we show that the maximum-depth separation problem is weakly N P-hard. We develop an exact pseudopolynomial-time dynamic programming algorithm by means of which we show that the maximum-depth separation problem is weakly N P-hard This result shows that, even if it entails (as we will show) the optimization of a hyperbolic objective function, the complexity class of this separation problem is the same as for the standard one based on the maximumviolation criterion. Our experiments reveal that a reduction in the number of cutting planes, inferior but similar to the one achieved with the maximumdepth criterion, can be obtained by separating maximum-violation inequalities and making the corresponding covers minimal via an a posteriori procedure. Computational experiments (see Sect. 8) on several classes of PHD-KP instances show that our dynamic-programming algorithm can solve the problem more efficiently than a state-of-the-art nonlinear-programming solver

On the maximum-violation cover inequalities

On the maximum-depth cover inequalities

On the nature of cover inequalities of maximum-depth

The values of the recursive function f are shown in

The point-to-hyperplane distance knapsack problem

Experiments when optimizing over the cover-inequality closure

Conclusions