Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Concurrency and Computation Practice and Experience

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

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

Published Nov 20, 2023• 2 citations
Publisher

Abstract

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.

DOI
10.1002/cpe.7967

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

Generating hard quadratic unconstrained binary optimization instances via the method of combining bit reduction and duplication technique

X. H. Li, Koji Nakano, Shunsuke Tsukiyama +3 more

Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization problem defined by an energy function that consists of a quadratic formula involving multiple binary variables. We propose the method of combing bit reduction and duplication technique that can generate hard instances of QUBO problems from any original QUBO problem without changing the size. The idea is to reduce the original QUBO problem for a specified number of bits and duplicate the same number of bits while ensuring all duplicated pair of bits take the same binary values. Experimental results show that generated QUBO instances are much hard for solving.

International Journal of Parallel Emergent and Distributed Systems•Jul 9, 2024•2 citations

Various QUBO Formulations of the Graph Isomorphism Problem and Related Problems

Daisuke Takafuji, 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 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.

Bulletin of Networking, Computing, Systems, and Software•Mar 1, 2024•0 citations

A benchmark QUBO problem inspired by digital halftoning based on the human visual system

Koji Nakano, Yasuaki Ito, Daisuke Takafuji +5 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. As many NP problems can be reduced to QUBO problems, considerable research has gone into developing QUBO solvers running on various computing platforms such as quantum devices, ASICs, FPGAs, GPUs, and optical fibers. For evaluating the performance of QUBO solvers, QUBO problems reduced from combinatorial optimization problems such as MaxCut, Traveling Salesman Problem (TSP), and Quadratic Assignment Problem (QAP) are commonly used. This paper presents a new benchmark QUBO problem inspired by digital halftoning, which aims to find a binary image reproducing an input grayscale image. The characteristics of this QUBO problem reduced from digital halftoning are quite unique. Solutions of the QUBO problem can be converted into binary images, from which we can visually evaluate the solution quality. Good approximation solutions for the QUBO problem can be obtained by a simple heuristic algorithm very quickly. However, it is very hard to find an optimal solution. Moreover, quite interestingly, we can create a QUBO problem for digital halftoning with a known optimal solution for a benchmarking purpose. Thus, we can exactly compute the gap between the optimal solution and an approximate solution obtained by a QUBO solver. We have developed an OpenMP-based sparse QUBO solver that can find good approximate solutions for QUBO problems reduced from digital halftoning. We have evaluated the performance of QUBO solvers including the D-Wave Hybrid solver, Gurobi optimizer, Open.Jij, and our sparse QUBO solver. The experimental results show that our OpenMP-based sparse QUBO solver attains the best performance.

International Symposium on Computing and Networking•Nov 1, 2022•1 citation

High‐throughput FPGA implementation for quadratic unconstrained binary optimization

Hiroshi Kagawa, Yasuaki Ito, Koji Nakano +6 more

Abstract Quadratic unconstrained binary optimization (QUBO) is a combinatorial optimization problem. Since various NP‐hard problems such as the traveling salesman problem can be formulated as a QUBO instance, QUBO is used with a wide range of applications. The main contribution of this article is to propose high‐throughput FPGA implementations for the QUBO solver. We perform the local search using different bit‐selection strategies based on the simulated annealing in the proposed implementation. The hardware is a pipeline structure with no pipeline hazards using multiple instances, where the bit‐flip operation is always performed every clock cycle. We implemented the proposed circuit on Xilinx UltraScale+ FPGA VU9P. The implementation result shows that the circuit can search solutions per second. Besides, by sharing the block RAM that stores a weight matrix, we implemented a dual annealer architecture that has two QUBO solvers into the FPGA. As a result, the dual annealer architecture can search solutions per second.

Concurrency and Computation Practice and Experience•Aug 19, 2021•15 citations

A Hybrid Framework Using a QUBO Solver For Permutation-Based Combinatorial Optimization

Siong Thye Goh, Sabrish Gopalakrishnan, Jianyuan Bo +1 more

In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine learning approach to tune the parameters for good performance effectively. To handle possible infeasibility, we introduce a polynomial-time projection algorithm. Finally, to solve large-scale problems, we introduce a divide-and-conquer approach that calls the QUBO solver repeatedly on small sub-problems. We tested our approach on provably hard Euclidean Traveling Salesman (E-TSP) instances and Flow Shop Problem (FSP). Optimality gap that is less than $10\%$ and $11\%$ are obtained respectively compared to the best-known approach.

arXiv:2009.12767•Sep 27, 2020•4 citations
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.
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.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization problem defined by an energy function that consists of a quadratic formula involving multiple binary variables. We propose the method of combing bit reduction and duplication technique that can generate hard instances of QUBO problems from any original QUBO problem without changing the size. The idea is to reduce the original QUBO problem for a specified number of bits and duplicate the same number of bits while ensuring all duplicated pair of bits take the same binary values. Experimental results show that generated QUBO instances are much hard for solving.
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. As many NP problems can be reduced to QUBO problems, considerable research has gone into developing QUBO solvers running on various computing platforms such as quantum devices, ASICs, FPGAs, GPUs, and optical fibers. For evaluating the performance of QUBO solvers, QUBO problems reduced from combinatorial optimization problems such as MaxCut, Traveling Salesman Problem (TSP), and Quadratic Assignment Problem (QAP) are commonly used. This paper presents a new benchmark QUBO problem inspired by digital halftoning, which aims to find a binary image reproducing an input grayscale image. The characteristics of this QUBO problem reduced from digital halftoning are quite unique. Solutions of the QUBO problem can be converted into binary images, from which we can visually evaluate the solution quality. Good approximation solutions for the QUBO problem can be obtained by a simple heuristic algorithm very quickly. However, it is very hard to find an optimal solution. Moreover, quite interestingly, we can create a QUBO problem for digital halftoning with a known optimal solution for a benchmarking purpose. Thus, we can exactly compute the gap between the optimal solution and an approximate solution obtained by a QUBO solver. We have developed an OpenMP-based sparse QUBO solver that can find good approximate solutions for QUBO problems reduced from digital halftoning. We have evaluated the performance of QUBO solvers including the D-Wave Hybrid solver, Gurobi optimizer, Open.Jij, and our sparse QUBO solver. The experimental results show that our OpenMP-based sparse QUBO solver attains the best performance.
Abstract Quadratic unconstrained binary optimization (QUBO) is a combinatorial optimization problem. Since various NP‐hard problems such as the traveling salesman problem can be formulated as a QUBO instance, QUBO is used with a wide range of applications. The main contribution of this article is to propose high‐throughput FPGA implementations for the QUBO solver. We perform the local search using different bit‐selection strategies based on the simulated annealing in the proposed implementation. The hardware is a pipeline structure with no pipeline hazards using multiple instances, where the bit‐flip operation is always performed every clock cycle. We implemented the proposed circuit on Xilinx UltraScale+ FPGA VU9P. The implementation result shows that the circuit can search solutions per second. Besides, by sharing the block RAM that stores a weight matrix, we implemented a dual annealer architecture that has two QUBO solvers into the FPGA. As a result, the dual annealer architecture can search solutions per second.
In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine learning approach to tune the parameters for good performance effectively. To handle possible infeasibility, we introduce a polynomial-time projection algorithm. Finally, to solve large-scale problems, we introduce a divide-and-conquer approach that calls the QUBO solver repeatedly on small sub-problems. We tested our approach on provably hard Euclidean Traveling Salesman (E-TSP) instances and Flow Shop Problem (FSP). Optimality gap that is less than 10%10\%10% and 11%11\%11% are obtained respectively compared to the best-known approach.