Faster exact solution of sparse MaxCut and QUBO problems

The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the...

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Published in:	Mathematical programming computation Vol. 15; no. 3; pp. 445 - 470
Main Authors:	Rehfeldt, Daniel, Koch, Thorsten, Shinano, Yuji
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023 Springer Nature B.V
Subjects:	Algorithms Combinatorial analysis Exact solutions Full Length Paper Mathematical analysis Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Operations Research/Decision Theory Optimization Quantum computers Solvers Theory of Computation 90C10 90C27 Maximum-cut Branch-and-cut Quadratic unconstrained binary optimization
ISSN:	1867-2949, 1867-2957
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Abstract	The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems—by using mathematical programming techniques. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse maximum-cut and quadratic unconstrained binary optimization instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.
AbstractList	The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems—by using mathematical programming techniques. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse maximum-cut and quadratic unconstrained binary optimization instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.
Author	Rehfeldt, Daniel Shinano, Yuji Koch, Thorsten
Author_xml	– sequence: 1 givenname: Daniel surname: Rehfeldt fullname: Rehfeldt, Daniel email: rehfeldt@zib.de organization: Applied Algorithmic Intelligence Methods departement, Zuse Institute Berlin – sequence: 2 givenname: Thorsten surname: Koch fullname: Koch, Thorsten organization: Applied Algorithmic Intelligence Methods departement, Zuse Institute Berlin, TU Berlin, Chair of Software and Algorithms for Discrete Optimization – sequence: 3 givenname: Yuji surname: Shinano fullname: Shinano, Yuji organization: Applied Algorithmic Intelligence Methods departement, Zuse Institute Berlin
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In: Ph.D. Thesis, University of Cologne (2004) – reference: DunningIGuptaSSilberholzJWhat works best when? A systematic evaluation of heuristics for max-cut and QUBOINFORMS J. Comput.201830360862410.1287/ijoc.2017.079807277784 – reference: JüngerMLobeEMutzelPReineltGRendlFRinaldiGStollenwerkTQuantum annealing versus digital computing: an experimental comparisonACM J. Exp. Algorithmics2021428615310.1145/34596061499.68123 – reference: HrgaTPovhJMADAM: a parallel exact solver for max-cut based on semidefinite programming and ADMMComput. Optim. Appl.2021802347375431747610.1007/s10589-021-00310-61478.90079 – reference: Wiegele, A.: BiqMac Library: A collection of Max-Cut and quadratic 0-1 programming instances of medium size. Tech. Rep. (2007) – reference: BonatoTJüngerMReineltGRinaldiGLifting and separation procedures for the cut polytopeMath. 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Snippet	The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the...
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SubjectTerms	Algorithms Combinatorial analysis Exact solutions Full Length Paper Mathematical analysis Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Operations Research/Decision Theory Optimization Quantum computers Solvers Theory of Computation
Title	Faster exact solution of sparse MaxCut and QUBO problems
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