Mathematical Programming

, Volume 171, Issue 1–2, pp 331–359 | Cite as

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

  • Merve BodurEmail author
  • Alberto Del Pia
  • Santanu S. Dey
  • Marco Molinaro
  • Sebastian Pokutta
Full Length Paper Series A


In this paper, we study the strength of Chvátal–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.


Integer programming Cutting planes Packing Covering Aggregation 

Mathematics Subject Classification




Santanu S. Dey would like to acknowledge the support of the NSF Grant CMMI#1149400 and Sebastian Pokutta would like to acknowledge the support of the NSF CAREER Award CMMI-1452463. Marco Molinaro would like to acknowledge the support of the grant CNPq Universal #431480/2016-8.


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Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada
  2. 2.Department of Industrial and Systems Engineering, Wisconsin Institute for DiscoveryUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  4. 4.Computer Science DepartmentPUC-RioRio de JaneiroBrazil

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