Designing faster mixed integer linear programming algorithm via learning the optimal path

TL;DR

The paper tackles slow MILP solving due to brittle heuristic node selection in branch-and-bound. It presents DeepBound, a deep learning-based node selector that uses a multi-level feature fusion network and a pairwise ranking objective to address severe data imbalance between oracle and non-oracle nodes. On three NP-hard MILP benchmarks, DeepBound achieves faster solving and earlier discovery of optimal nodes, with strong generalization to larger instances, outperforming BES and prior ML approaches. The work demonstrates that problem-specific feature learning can reveal robust node-selection patterns and potentially reduce reliance on handcrafted heuristics in MILP solvers.

Abstract

Designing faster algorithms for solving Mixed-Integer Linear Programming (MILP) problems is highly desired across numerous practical domains, as a vast array of complex real-world challenges can be effectively modeled as MILP formulations. Solving these problems typically employs the branch-and-bound algorithm, the core of which can be conceived as searching for a path of nodes (or sub-problems) that contains the optimal solution to the original MILP problem. Traditional approaches to finding this path rely heavily on hand-crafted, intuition-based heuristic strategies, which often suffer from unstable and unpredictable performance across different MILP problem instances. To address this limitation, we introduce DeepBound, a deep learning-based node selection algorithm that automates the learning of such human intuition from data. The core of DeepBound lies in learning to prioritize nodes containing the optimal solution, thereby improving solving efficiency. DeepBound introduces a multi-level feature fusion network to capture the node representations. To tackle the inherent node imbalance in branch-and-bound trees, DeepBound employs a pairwise training paradigm that enhances the model's ability to discriminate between nodes. Extensive experiments on three NP-hard MILP benchmarks demonstrate that DeepBound achieves superior solving efficiency over conventional heuristic rules and existing learning-based approaches, obtaining optimal feasible solutions with significantly reduced computation time. Moreover, DeepBound demonstrates strong generalization capability on large and complex instances. The analysis of its learned features reveals that the method can automatically discover more flexible and robust feature selection, which may effectively improve and potentially replace human-designed heuristic rules.

Figures (10)

• Figure 1: MILP problems and branch-and-bound tree. a, The general form of MILP problem. b, Three types of NP-Hard MILP problem included in our testing dataset. c, A complex large-scale set covering problem in MILP form. The matrices $A$, $b$, and $c^{T}$ in this example are highlighted in corresponding colors. d, Part of the branch-and-bound tree obtained by solving the set covering problem in c. Path to the optimal solution are marked in light red. Statistical analysis reveals a severe imbalance in different node types.
• Figure 2: DeepBound architecture. Overview of using DeepBound for node selection, sorting nodes based on their scores and iteratively selecting the higher-ranked node each time to find the optimal solution. DeepBound utilizes the ensemble learning mechanism of fusion models to process feature vectors of node pairs and provides average score for node ranking. Each fusion block performs node-level and feature-level fusion on the feature vectors of node pairs for node comparison and subsequent node scoring.
• Figure 3: Evaluation of DeepBound node selector with MILP test sets of different types and scales.a-c, Evaluation on instances of three different MILP problems. The figures show the distribution of solving time and bpb time for the different node selection methods on 100 randomly generated instances of each type of MILP problem, with the 95% confidence ellipses. d-f, Evaluation of the convergence pattern of primal gap for different node selection methods in solving three different MILP problems, with the mean curves and 95% confidence intervals. g-i, Comparison of the number of different node selection methods that achieve the fastest solving speed (wins) on the same set of 100 large-scale hard problem instances for each type of MILP problem.
• Figure 4: DeepBound accelerates the solving of MILP problems. Comparison of the branch-and-bound trees and the node-primal gap curves between DeepBound and BES when solving identical 2000$\times$1000 set covering problems. The central panel illustrates the node-primal gap curves, which depict the relationship between the number of already explored nodes and the primal gap at the node. Figures on either side visualize the branch-and-bound tree of DeepBound and BES at different stages of the solving process. The oracle node paths leading to the discovery of the optimal feasible solution within each tree are emphasized in light red shading.
• Figure 5: Evaluation of the distance to the optimal solution.a-c, Evaluation of the decreasing trend of the distance for different node selection methods between the current node and the node containing the optimal solution in the branch-and-bound tree before the optimal solution was found. The mean curves and 95% confidence intervals are depicted.