Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

Authors Michael Jünger, Sven Mallach

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Michael Jünger
  • University of Cologne, Germany
Sven Mallach
  • University of Cologne, Germany

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Michael Jünger and Sven Mallach. Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Solving the NP-hard Maximum Cut or Binary Quadratic Optimization Problem to optimality is important in many applications including Physics, Chemistry, Neuroscience, and Circuit Layout. The leading approaches based on linear/semidefinite programming require the separation of so-called odd-cycle inequalities for solving relaxations within their associated branch-and-cut frameworks. In their groundbreaking work, F. Barahona and A.R. Mahjoub have given an informal description of a polynomial-time separation procedure for the odd-cycle inequalities. Since then, the odd-cycle separation problem has broadly been considered solved. However, as we reveal, a straightforward implementation is likely to generate inequalities that are not facet-defining and have further undesired properties. Here, we present a more detailed analysis, along with enhancements to overcome the associated issues efficiently. In a corresponding experimental study, it turns out that these are worthwhile, and may speed up the solution process significantly.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Linear programming
  • Mathematics of computing → Integer programming
  • Maximum cut
  • Binary quadratic optimization
  • Integer linear programming


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