An abstract model for branch and cut

Aleksandr M. Kazachkov, Pierre Le Bodic, Sriram Sankaranarayanan

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Abstract

Branch and cut is the dominant paradigm for solving a wide range of mathematical programming problems—linear or nonlinear—combining efficient search (via branch and bound) and relaxation-tightening procedures (via cutting planes, or cuts). While there is a wealth of computational experience behind existing cutting strategies, there is simultaneously a relative lack of theoretical explanations for these choices, and for the tradeoffs involved therein. Recent papers have explored abstract models for branching and for comparing cuts with branch and bound. However, to model practice, it is crucial to understand the impact of jointly considering branching and cutting decisions. In this paper, we provide a framework for analyzing how cuts affect the size of branch-and-cut trees, as well as their impact on solution time. Our abstract model captures some of the key characteristics of real-world phenomena in branch-and-cut experiments, regarding whether to generate cuts only at the root or throughout the tree, how many rounds of cuts to add before starting to branch, and why cuts seem to exhibit nonmonotonic effects on the solution process.

Original languageEnglish
Number of pages28
JournalMathematical Programming
DOIs
Publication statusAccepted/In press - 30 Jun 2023

Keywords

  • Branch and bound
  • Cutting planes
  • Integer programming
  • An abstract model for branch-and-cut

    Kazachkov, A. M., Le Bodic, P. & Sankaranarayanan, S., 2022, Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022 Eindhoven, The Netherlands, June 27–29, 2022 Proceedings. Aardal, K. & Sanità, L. (eds.). Cham Switzerland: Springer, p. 333-346 14 p. (Lecture Notes in Computer Science; vol. 13265).

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