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Learning in Local Branching (Invited Talk)

Authors Defeng Liu, Andrea Lodi

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Author Details

Defeng Liu
  • CERC, Polytechnique Montréal, Canada
Andrea Lodi
  • Jacobs Technion-Cornell Institute, Cornell Tech and Technion - Israel Institute of Technology, New York, NY, USA
  • CERC, Polytechnique Montréal, Canada


The authors are indebted to Matteo Fischetti for several conversations about the work in pregress.

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Defeng Liu and Andrea Lodi. Learning in Local Branching (Invited Talk). In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 3:1-3:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Although state-of-the-art solvers for Mixed-Integer Programming (MIP) experienced a dramatic performance improvement over the past decades, the resolution of some MIPs is still challenging, requiring hours of computations while, in practice, high-quality solutions are often required to be computed within a very restricted time frame. In such cases, it might be preferable to provide anytime solutions, i.e., a first reasonable solution should be generated as early as possible, then better ones produced in the subsequent computation with the user deciding where to stop. In this respect, the local branching (LB) heuristic [Fischetti and Lodi, 2003] was proposed to improve an incumbent solution either at very early stages of the computation within a general MIP framework or as a stand-alone algorithmic framework. Roughly speaking, given a feasible solution, the method iterates by first defining a solution neighborhood through the so-called local branching cut, then by exploring it by calling a black-box MIP solver. In the local branching algorithm, the choice of the neighborhood size is crucial to performance. In principle, it is desirable to have neighborhoods to be relatively small for efficient computation but still large enough to contain improving solutions. In [Fischetti and Lodi, 2003], the size of the neighborhood is mostly initialized by a fixed constant value, then adjusted at run time. Nonetheless, it is reasonable to believe that there is no a priori single best neighborhood size and the choice of the value should depend on the characteristics of the problem. Furthermore, it is worth noting that, in many applications, instances of the same problem are solved repeatedly. Real-world problems have a rich structure: while more and more data points are collected, patterns and regularities appear. Therefore, problem-specific and task-specific knowledge can be learned from data and applied to adapting the corresponding optimization scenario. This motives a broader paradigm of sizing the solution neighborhoods in local branching. Following the line of work analyzed and surveyed in [Bengio et al., 2021] on the use of Machine Learning (ML) for combinatorial optimization, in this work, we aim to guide the (local) search of the local branching heuristic by ML techniques. In particular, given a problem instance and a time limit for (heuristically) solving it, we exploit ML tools to predict reasonable good values of the neighborhood size, in order to maximize the performance of the local branching algorithm. We computationally show that the neighborhood size can indeed be learnt leading to improved performances and that the overall algorithm generalizes well both with respect to the instance size and, more surprisingly, across instances.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Machine learning
  • Mathematics of computing → Discrete optimization
  • Local search
  • learning
  • mixed-integer programming


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  1. Yoshua Bengio, Andrea Lodi, and Antoine Prouvost. Machine learning for combinatorial optimization: A methodological tour d'horizon. European Journal of Operational Research, 290(2):405-421, 2021. Google Scholar
  2. Matteo Fischetti and Andrea Lodi. Local branching. Mathematical programming, 98(1-3):23-47, 2003. Google Scholar
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