Aggregation-based cutting-planes for packing and covering integer programs

TL;DR

This paper investigates the strength of aggregation-based cutting planes for packing and covering integer programs, showing that simple non-aggregated cuts can approximate aggregation closures and exploring the power of multi-row cuts.

Contribution

It demonstrates that for packing and covering IPs, aggregation closures can be approximated by non-aggregated cuts, and analyzes the effectiveness of multi-row cuts in relation to the integrality gap.

Findings

Aggregation cuts can be arbitrarily stronger than individual constraint cuts.

For packing and covering IPs, the CG and aggregation closures are 2-approximable by non-aggregated cuts.

Large integrality gaps imply high k-aggregation closure rank.

Abstract

In this paper, we study the strength of Chvatal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.