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Bulletin of Networking, Computing, Systems, and Software

Various QUBO Formulations of the Graph Isomorphism Problem and Related Problems

Daisuke Takafuji, Koji Nakano, Yasuaki Ito, Takashi Yazane, Junko Yano, Takumi Kato, Shiro Ozaki, Rie Mori, Ryota Katsuki

Published Mar 1, 2024• 0 citations
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Abstract

Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The graph isomorphism problem is one of the applications of combinatorial optimization, and is not known to be solvable in polynomial time. The problem is also not known to be NP-complete. We propose four QUBO formulations for graph isomorphism problem instances. We also propose four QUBO formulations for induced subgraph isomorphism problem instances and two QUBO formulations for subgraph isomorphism problem instances, which are NP-complete problems. We solve QUBO instances defined by our QUBO formulations, using known QUBO solvers: Gurobi optimizer, Fixstars Amplify AE and OpenJij with SA.

Related papers

Bit duplication technique to generate hard QUBO problems

Xiaotian Li, Koji Nakano, Yasuaki Ito +6 more

Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this paper is to present the bit duplication technique, which can generate a hard QUBO problem from any QUBO problem so that their optimal solutions take the same energy value. The idea of the bit duplication technique is to duplicate bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with$n$bits is converted to a hard QUBO problem with$2n$bits. We use random QUBO problems and N-Queen QUBO problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D-Wave Hybrid solver, and ABS QUBO solver are evaluated for solving these QUBO problems. The experimental results show that generated QUBO problems by the bit duplication technique are much harder than the original QUBO problems. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.

International Symposium on Computing and Networking Workshops (CANDARW)•Nov 1, 2022•2 citations

Uma reformulação quadrática para variáveis binárias aplicadas ao problema de otimização quadrática irrestrita

Petra Maria Bartmeyer, Christiano Lyra

The Quadratic Unconstrained Binary Optimization problem (QUBO) is the optimization of a quadratic function where the variables are binary; it is frequently used to model combinatorial problems in operational research and graph theory.QUBO is an NP-Complete problem and a challenge to exact methods.Therefore, the problem has been approached with heuristics.Here, we propose new ideas to solve QUBO with exact methods.We develop a new quadratic formulation that maps the binary problem into an equivalent continuous optimization problem with a single quadratic constraint.A further development delves into Lagrangian inspired concepts to obtain an equivalent unconstrained formulation.The case studies illustrate the benefit and limitations of both formulations.Also, the computational experiments show that problems with up to 2500 variables were solved with processing times under 500 seconds to achieve an average gap under 1%; these results also show that the approach allowed to solve, with an exact method, instances of 1500 more binary variables than the instances considered in previous papers.

Anais do Simpósio Brasileiro de Pesquisa Operacional•Jan 1, 2018•0 citations

QUBO formulations of three NP problems

Anuradha Mahasinghe, Vishmi Fernando, Paduma Samarawickrama

With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.

Journal of Information and Optimization Sciences•Oct 3, 2021•6 citations

Bit duplication technique to generate hard quadratic unconstrained binary optimization problems with adjustable sizes

X. H. Li, Koji Nakano, Yasuaki Ito +7 more

Summary Quadratic unconstrained binary optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this article is to propose the bit duplication technique that can specify the number of duplicated bits, so that it can generate hard QUBO problem with adjustable sizes. The idea is to duplicate specified number of bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with bits is converted to a hard QUBO problem with bits . We use random QUBO problems, N‐Queen problems, traveling salesman problem and maximum weight matching problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D‐Wave samplers with SA, D‐Wave hybrid and ABS2 QUBO solver are evaluated for solving these QUBO problems. The experimental results show that only a small scale of duplicated bits can make QUBO problems harder. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.

Concurrency and Computation Practice and Experience•Nov 20, 2023•2 citations

QUBO++: A C++ Library for Developing and Solving QUBO Problems

Koji Nakano, Shunsuke Tsukiyama, Xiaotian Li +5 more

The Quadratic Unconstrained Binary Optimization (QUBO) model is widely used to solve combinatorial optimization problems by determining the optimal assignment of binary variables in a quadratic formulation. Given its versatility, extensive research has been conducted on developing QUBO solvers and converting combinatorial optimization problems into equivalent QUBO formulations. This paper introduces QUBO++, a novel C++ symbolic mathematical computation library specifically designed for QUBO modeling. It provides a comprehensive set of operators and functions for symbolic computation, enabling seamless transformation of optimization problems into QUBO models. While several Python-based libraries exist for similar purposes, QUBO++ is the only dedicated C++ library that supports multithreaded acceleration via Intel Threading Building Blocks (oneTBB). In addition to its modeling capabilities, QUBO++ includes two fundamental QUBO solvers: the Exhaustive Solver, which is useful for verifying small-scale problems, and the Easy Solver, designed for efficient testing and debugging. Compared to existing Python-based QUBO modeling tools such as PyQUBO, Fixstars Amplify, and Tytan, QUBO++ demonstrates significantly higher performance, achieving speedups of over 1,700 times in certain scenarios. By leveraging QUBO-based problem-solving techniques, QUBO++ serves as a powerful and versatile tool for addressing complex optimization challenges in both industrial and societal applications.

IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum•Jun 3, 2025•3 citations

Graph-theoretic Formulation of QUBO for Scalable Local Search on GPUs

Ryota Yasudo, Koji Nakano, Yasuaki Ito +5 more

Heuristic solvers of the Quadratic Unconstrained Binary Optimization (QUBO) problems have been rapidly studied because of their potentialities to solve a wide range of com-binatorial optimization problems in a unified manner. QUBO solvers, however, face a limitation in the numbers of variables. To overcome this limitation, we reformulate QUBO as a maximum weight induced subgraph problem (MISG) and propose a graph-theoretic approach for solving MISG. It is a problem to find an induced subgraph with the maximum total weight for a given edge-weighted undirected graph. Naive local search that updates only one vertex at a time requires many iterations to obtain good solutions for large-scale problems. To accelerate it, we introduce a new technique to update multiple vertices that can improve the solution at the same time. We have implemented a MISG solver based on our algorithm on NVIDIA A100 GPUs, where flip operations are parallelized with GPU multithreaded computing. Our solver is scalable and supports up to 230(1G) variables. Our results show that our algorithm outperforms the Gurobi optimizer in terms of solution qualities and the execution time.

IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum•May 1, 2022•6 citations
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The graph isomorphism problem is one of the applications of combinatorial optimization, and is not known to be solvable in polynomial time. The problem is also not known to be NP-complete. We propose four QUBO formulations for graph isomorphism problem instances. We also propose four QUBO formulations for induced subgraph isomorphism problem instances and two QUBO formulations for subgraph isomorphism problem instances, which are NP-complete problems. We solve QUBO instances defined by our QUBO formulations, using known QUBO solvers: Gurobi optimizer, Fixstars Amplify AE and OpenJij with SA.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this paper is to present the bit duplication technique, which can generate a hard QUBO problem from any QUBO problem so that their optimal solutions take the same energy value. The idea of the bit duplication technique is to duplicate bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem withnnnbits is converted to a hard QUBO problem with2n2n2nbits. We use random QUBO problems and N-Queen QUBO problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D-Wave Hybrid solver, and ABS QUBO solver are evaluated for solving these QUBO problems. The experimental results show that generated QUBO problems by the bit duplication technique are much harder than the original QUBO problems. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.
The Quadratic Unconstrained Binary Optimization problem (QUBO) is the optimization of a quadratic function where the variables are binary; it is frequently used to model combinatorial problems in operational research and graph theory.QUBO is an NP-Complete problem and a challenge to exact methods.Therefore, the problem has been approached with heuristics.Here, we propose new ideas to solve QUBO with exact methods.We develop a new quadratic formulation that maps the binary problem into an equivalent continuous optimization problem with a single quadratic constraint.A further development delves into Lagrangian inspired concepts to obtain an equivalent unconstrained formulation.The case studies illustrate the benefit and limitations of both formulations.Also, the computational experiments show that problems with up to 2500 variables were solved with processing times under 500 seconds to achieve an average gap under 1%; these results also show that the approach allowed to solve, with an exact method, instances of 1500 more binary variables than the instances considered in previous papers.
With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.
Summary Quadratic unconstrained binary optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this article is to propose the bit duplication technique that can specify the number of duplicated bits, so that it can generate hard QUBO problem with adjustable sizes. The idea is to duplicate specified number of bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with bits is converted to a hard QUBO problem with bits . We use random QUBO problems, N‐Queen problems, traveling salesman problem and maximum weight matching problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D‐Wave samplers with SA, D‐Wave hybrid and ABS2 QUBO solver are evaluated for solving these QUBO problems. The experimental results show that only a small scale of duplicated bits can make QUBO problems harder. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.
The Quadratic Unconstrained Binary Optimization (QUBO) model is widely used to solve combinatorial optimization problems by determining the optimal assignment of binary variables in a quadratic formulation. Given its versatility, extensive research has been conducted on developing QUBO solvers and converting combinatorial optimization problems into equivalent QUBO formulations. This paper introduces QUBO++, a novel C++ symbolic mathematical computation library specifically designed for QUBO modeling. It provides a comprehensive set of operators and functions for symbolic computation, enabling seamless transformation of optimization problems into QUBO models. While several Python-based libraries exist for similar purposes, QUBO++ is the only dedicated C++ library that supports multithreaded acceleration via Intel Threading Building Blocks (oneTBB). In addition to its modeling capabilities, QUBO++ includes two fundamental QUBO solvers: the Exhaustive Solver, which is useful for verifying small-scale problems, and the Easy Solver, designed for efficient testing and debugging. Compared to existing Python-based QUBO modeling tools such as PyQUBO, Fixstars Amplify, and Tytan, QUBO++ demonstrates significantly higher performance, achieving speedups of over 1,700 times in certain scenarios. By leveraging QUBO-based problem-solving techniques, QUBO++ serves as a powerful and versatile tool for addressing complex optimization challenges in both industrial and societal applications.
Heuristic solvers of the Quadratic Unconstrained Binary Optimization (QUBO) problems have been rapidly studied because of their potentialities to solve a wide range of com-binatorial optimization problems in a unified manner. QUBO solvers, however, face a limitation in the numbers of variables. To overcome this limitation, we reformulate QUBO as a maximum weight induced subgraph problem (MISG) and propose a graph-theoretic approach for solving MISG. It is a problem to find an induced subgraph with the maximum total weight for a given edge-weighted undirected graph. Naive local search that updates only one vertex at a time requires many iterations to obtain good solutions for large-scale problems. To accelerate it, we introduce a new technique to update multiple vertices that can improve the solution at the same time. We have implemented a MISG solver based on our algorithm on NVIDIA A100 GPUs, where flip operations are parallelized with GPU multithreaded computing. Our solver is scalable and supports up to 230(1G) variables. Our results show that our algorithm outperforms the Gurobi optimizer in terms of solution qualities and the execution time.