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Cut-based Conflict Analysis in Mixed Integer Programming

Gioni Mexi, Felipe Serrano, Timo Berthold, Ambros Gleixner, Jakob Nordström

TL;DR

This work reframes MIP conflict analysis from a graph-based to a cut-based paradigm, treating conflict learning as a sequence of linear combinations, integer roundings, and cuts derived from linear constraints. It introduces a new mixed-integer rounding (MIR) based reduction for pure binary programs and shows dominance over prior reduction strategies, including CG and previous MIR approaches. The authors extend cut-based conflict analysis to mixed binary programs by incorporating non-relaxable continuous propagations, and propose a MBP-specific reduction to handle continuous variables while preserving tight propagation for binary decisions. Implemented in SCIP and evaluated on MIPLIB2017, the cut-based methods improve running time, reduce search tree size, and solve more instances compared to graph-based conflict analysis, though general MIP with arbitrary integer variables remains challenging with a fallback to traditional methods. Overall, the results demonstrate that cut-based conflict analysis can meaningfully augment MIP solvers, offering robust improvements in practice and insights into the interaction between cuts, reductions, and propagation within conflict learning.

Abstract

For almost two decades, mixed integer programming (MIP) solvers have used graph-based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this paper, we improve MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo-Boolean optimization. Phrased in MIP terminology, this type of conflict analysis can be understood as a sequence of linear combinations, integer roundings, and cut generation. We leverage this MIP perspective to design a new conflict analysis algorithm based on mixed integer rounding cuts, which theoretically dominates the state-of-the-art method in pseudo-Boolean optimization using Chvátal-Gomory cuts. Furthermore, we extend this cut-based conflict analysis from pure binary programs to mixed binary programs and-in limited form-to general MIP with also integer-valued variables. We perform an empirical evaluation of cut-based conflict analysis as implemented in the open-source MIP solver SCIP, testing it on a large and diverse set of MIP instances from MIPLIB 2017. Our experimental results indicate that the new algorithm improves the default performance of SCIP in terms of running time, number of nodes in the search tree, and the number of instances solved.

Cut-based Conflict Analysis in Mixed Integer Programming

TL;DR

This work reframes MIP conflict analysis from a graph-based to a cut-based paradigm, treating conflict learning as a sequence of linear combinations, integer roundings, and cuts derived from linear constraints. It introduces a new mixed-integer rounding (MIR) based reduction for pure binary programs and shows dominance over prior reduction strategies, including CG and previous MIR approaches. The authors extend cut-based conflict analysis to mixed binary programs by incorporating non-relaxable continuous propagations, and propose a MBP-specific reduction to handle continuous variables while preserving tight propagation for binary decisions. Implemented in SCIP and evaluated on MIPLIB2017, the cut-based methods improve running time, reduce search tree size, and solve more instances compared to graph-based conflict analysis, though general MIP with arbitrary integer variables remains challenging with a fallback to traditional methods. Overall, the results demonstrate that cut-based conflict analysis can meaningfully augment MIP solvers, offering robust improvements in practice and insights into the interaction between cuts, reductions, and propagation within conflict learning.

Abstract

For almost two decades, mixed integer programming (MIP) solvers have used graph-based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this paper, we improve MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo-Boolean optimization. Phrased in MIP terminology, this type of conflict analysis can be understood as a sequence of linear combinations, integer roundings, and cut generation. We leverage this MIP perspective to design a new conflict analysis algorithm based on mixed integer rounding cuts, which theoretically dominates the state-of-the-art method in pseudo-Boolean optimization using Chvátal-Gomory cuts. Furthermore, we extend this cut-based conflict analysis from pure binary programs to mixed binary programs and-in limited form-to general MIP with also integer-valued variables. We perform an empirical evaluation of cut-based conflict analysis as implemented in the open-source MIP solver SCIP, testing it on a large and diverse set of MIP instances from MIPLIB 2017. Our experimental results indicate that the new algorithm improves the default performance of SCIP in terms of running time, number of nodes in the search tree, and the number of instances solved.

Paper Structure

This paper contains 21 sections, 4 theorems, 41 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Questions Studied in This Work and Our Contributions
  3. Organization of This Paper
  4. Preliminaries and Notation
  5. States and Activities
  6. Propagation of Linear Constraints
  7. Coefficient Tightening and Cutting Planes
  8. Weakening and Complementation
  9. The Simple Case: Conflict Analysis under Tight Propagations
  10. Reduction Techniques for Pure Binary Programs
  11. Coefficient Tightening-based Reduction
  12. cMIR-based Reduction
  13. Dominance Relationships
  14. Conflict Analysis for Mixed Binary Programs
  15. Reduction Algorithm for Mixed Binary Programs
  16. ...and 6 more sections

Key Result

Proposition 1

Let $C_{\textnormal{reason}}: a_r x_{r}+ \sum_{ j \neq r } a_j x_{j} \geq b$ be a constraint propagating a variable $x_{r}$ in the state $\rho_{}$ tightly. Further, assume that $C_{\textnormal{confl}}: a'_{r}x_{r} + \sum_{ j \neq r } a'_{j}x_{j} \geq b'$ becomes infeasible in the state $\rho_{}$. Th

Figures (3)

  • Figure 1: Inequality $x_1 + x_2 + 2x_3 \geq 2$ on the face of the polytope with $x_1 = 0$. The fractional vertex is $(0, 1, 0.5)$.
  • Figure 2: Left: Region defined by $C_2$, $C_3$, and the global domain; Middle: Mixed‐integer hull; Right: Mixed‐integer hull intersected with the local domain.
  • Figure 3: Example where it is impossible to find a linear cut that separates the non-integer vertex (2,0.5) from the feasible region.

Theorems & Definitions (18)

  • Definition 1: Coefficient Tightening
  • Definition 2: Chvátal-Gomory Cut
  • Definition 3: Mixed Integer Rounding Cut
  • Remark 1
  • Example 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • ...and 8 more