[Paper Review] Structural Analysis of Branch-and-Cut and the Learnability of Gomory Mixed Integer Cuts

Maria-Florina Balcan, Siddharth Prasad+2 more|arXiv (Cornell University)|Apr 15, 2022

Machine Learning and Algorithms4 citations

TL;DR

This paper provides a structural analysis of branch-and-cut algorithms, establishing sample complexity bounds for learning optimal Gomory mixed integer (GMI) cut parameters using machine learning. By characterizing how LP solutions and B&C search trees change with cut parameters through polynomial partitioning and geometric analysis, the authors derive pseudo-dimension bounds that enable generalization guarantees for cut selection across infinite families of cuts, including GMI cuts.

ABSTRACT

The incorporation of cutting planes within the branch-and-bound algorithm, known as branch-and-cut, forms the backbone of modern integer programming solvers. These solvers are the foremost method for solving discrete optimization problems and thus have a vast array of applications in machine learning, operations research, and many other fields. Choosing cutting planes effectively is a major research topic in the theory and practice of integer programming. We conduct a novel structural analysis of branch-and-cut that pins down how every step of the algorithm is affected by changes in the parameters defining the cutting planes added to the input integer program. Our main application of this analysis is to derive sample complexity guarantees for using machine learning to determine which cutting planes to apply during branch-and-cut. These guarantees apply to infinite families of cutting planes, such as the family of Gomory mixed integer cuts, which are responsible for the main breakthrough speedups of integer programming solvers. We exploit geometric and combinatorial structure of branch-and-cut in our analysis, which provides a key missing piece for the recent generalization theory of branch-and-cut.

Motivation & Objective

• To understand how changes in Gomory mixed integer cut parameters affect the entire branch-and-cut (B&C) search tree.
• To provide generalization guarantees for machine learning-based cut selection by analyzing the intrinsic complexity of B&C as a function of cut parameters.
• To bridge the gap in generalization theory for branch-and-cut by analyzing the interplay between continuous LP solution changes and discrete tree structure.
• To derive sample complexity bounds applicable to infinite families of cuts, such as GMI cuts, under distributional assumptions on integer programs.
• To formalize the impact of cut selection on B&C performance, showing that optimal parameters can vary drastically across different instance distributions.

Proposed method

• Partition the space of constraint vectors into regions where the LP solution has a closed-form expression, using boundaries defined by degree-2 polynomial inequalities.
• Analyze the variation of the B&C search tree as a function of cut parameters by tracking how LP solutions change across these regions.
• Use polynomial inequalities to describe the set of cut parameter vectors that yield the same B&C tree structure, with degrees bounded by $2dK$ for $d$-degree constraints.
• Apply Warren’s theorem and the Milnor-Thom theorem to bound the number of connected components in the parameter space where the B&C tree structure remains invariant.
• Establish a pseudo-dimension bound for the class of functions mapping cut parameters to the number of nodes in the B&C tree, using the number of polynomial surfaces and connected components.
• Leverage the main result of Balcan et al. [6] to derive a pseudo-dimension bound of $ O(mK^3\log U + mn^3K^2\log(mnK\tau) + mK^2\log(ab)) $ for the function class of B&C tree sizes.

Experimental results

Research questions

• RQ1 How does the solution to the LP relaxation change as new cutting planes are added, particularly in terms of discontinuous jumps between vertices?
• RQ2 How does the structure of the branch-and-cut search tree vary as a function of the parameters defining the cutting planes?
• RQ3 What is the intrinsic complexity of the mapping from cut parameters to B&C tree size, and how can it be bounded using geometric and combinatorial tools?
• RQ4 Can generalization guarantees be established for machine learning-based cut selection in branch-and-cut, and what sample complexity is required?
• RQ5 How do distributional differences in integer programs affect the optimal choice of cut parameters, and can this be captured formally?

Key findings

• The space of cut parameters can be partitioned into $ 2^{O(n^2)}K^{O(n^3)}(m+2n)^{O(n^3)}\tau^{O(n^3)} $ regions where the B&C tree structure remains invariant.
• The number of distinct B&C tree structures induced by $ K $ sequential GMI cuts is bounded by a pseudo-dimension of $ O(mK^3\log U + mn^3K^2\log(mnK\tau) + mK^2\log(ab)) $.
• The set of cut parameter vectors that yield the same B&C tree can be described by the intersection of $ O(12^{n(n+1)}n^{2n(n+1)}K^{2n^{2}(n+1)}(m+2n)^{2n^{2}(n+1)}\tau^{5n^{2}(n+1)}) $ polynomial inequalities of degree $ 4K^2 $.
• The analysis reveals that small changes in cut parameters can cause large changes in the B&C tree due to discontinuous jumps in LP solutions, necessitating careful structural modeling.
• The results show that optimal cut parameters can be fundamentally different across distributions—e.g., a smaller $ \mu $ is better in one distribution, a larger $ \mu $ in another—highlighting the need for distribution-aware cut learning.
• The derived bounds provide generalization guarantees for any cut selection method, whether optimal or suboptimal, over training sets drawn from an unknown distribution of IPs.