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Non-Monotonicity of Branching Rules with respect to Linear Relaxations

Prachi Shah, Santanu S. Dey, Marco Molinaro

TL;DR

It is proved that any branching rule which exclusively branches on fractional variables in the LP solution is nonmonotonic, as well as a family of instances where adding a single cut leads to an exponential increase in the size of full strong branching trees, despite improving the LP bound.

Abstract

Modern mixed-integer programming solvers use the branch-and-cut framework, where cutting planes are added to improve the tightness of the linear programming (LP) relaxation, with the expectation that the tighter formulation would produce smaller branch-and-bound trees. In this work, we consider the question of whether adding cuts will always lead to smaller trees for a given fixed branching rule. We formally call such a property of a branching rule monotonicity. We prove that any branching rule which exclusively branches on fractional variables in the LP solution is non-monotonic. Moreover, we present a family of instances where adding a single cut leads to an exponential increase in the size of full strong branching trees, despite improving the LP bound. Finally, we empirically attempt to estimate the prevalence of non-monotonicity in practice while using full strong branching. We consider randomly generated multi-dimensional knapsacks tightened by cover cuts as well as instances from the MIPLIB 2017 benchmark set for the computational experiments. Our main insight from these experiments is that if the gap closed by cuts is small, change in tree size is difficult to predict, and often increases, possibly due to inherent non-monotonicity. However, when a sufficiently large gap is closed, a significant decrease in tree size may be expected.

Non-Monotonicity of Branching Rules with respect to Linear Relaxations

TL;DR

It is proved that any branching rule which exclusively branches on fractional variables in the LP solution is nonmonotonic, as well as a family of instances where adding a single cut leads to an exponential increase in the size of full strong branching trees, despite improving the LP bound.

Abstract

Modern mixed-integer programming solvers use the branch-and-cut framework, where cutting planes are added to improve the tightness of the linear programming (LP) relaxation, with the expectation that the tighter formulation would produce smaller branch-and-bound trees. In this work, we consider the question of whether adding cuts will always lead to smaller trees for a given fixed branching rule. We formally call such a property of a branching rule monotonicity. We prove that any branching rule which exclusively branches on fractional variables in the LP solution is non-monotonic. Moreover, we present a family of instances where adding a single cut leads to an exponential increase in the size of full strong branching trees, despite improving the LP bound. Finally, we empirically attempt to estimate the prevalence of non-monotonicity in practice while using full strong branching. We consider randomly generated multi-dimensional knapsacks tightened by cover cuts as well as instances from the MIPLIB 2017 benchmark set for the computational experiments. Our main insight from these experiments is that if the gap closed by cuts is small, change in tree size is difficult to predict, and often increases, possibly due to inherent non-monotonicity. However, when a sufficiently large gap is closed, a significant decrease in tree size may be expected.
Paper Structure (19 sections, 2 theorems, 15 equations, 8 figures)

This paper contains 19 sections, 2 theorems, 15 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formalizing the size of a tree for a given branching rule.
  3. Our contributions.
  4. Non-Monotonicity of Branching Rules
  5. Background on branching rules
  6. Any rule that branches exclusively on fractional variables is non-monotonic
  7. Exponential increase in size of branch-and-bound tree
  8. Computational Experiments
  9. Cover Cuts for Multi-dimensional Knapsack Problem (MKP)
  10. Instances.
  11. Experiments.
  12. Results.
  13. Experiments on MIPLIB
  14. Instances.
  15. Experiments.
  16. ...and 4 more sections

Key Result

Theorem 1

Any branching rule that branches only on fractional variables in the optimal LP solution is non-monotonic.

Figures (8)

  • Figure 1: Illustration of branch-and-bound trees for the example in Theorem \ref{['theorem:frac_rules']}.
  • Figure 2:
  • Figure 3: Illustration of strong branching trees for the example in Theorem \ref{['theorem:exp_SB-P']}.
  • Figure 4: Impact of adding cover cuts on the size of FSB-P trees for MKP instances.
  • Figure 5: Progression of change in tree size and gap closed as the number of rounds of cuts is increased from 0 to 10.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1