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.
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 withnbits is converted to a hard QUBO problem with2nbits. 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.
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 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. 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.
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.
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.
A Quadratic Unconstrained Binary Optimization (QUBO) model is defined by a quadratic objective function comprising binary variables, each taking values of 0 or 1. The objective of the QUBO problem is to determine the optimal assignment of these binary variables to minimize the QUBO model. QUBO models are instrumental in solving various combinatorial optimization problems, leading to extensive research on solver development, as many combinatorial optimization problems can be reformulated as QUBO problems. Notably, D-Wave Systems, a company specializing in quantum technologies, has developed quantum annealers which are programmable QUBO solvers that utilize quantum mechanics. SHA-1 is a cryptographic hash function that generates a 160-bit hash value from a message of an arbitrary length. This hash value makes it difficult to compute the original message within a practical timeframe. A pre-image attack, which aims to compute the original message from a hash value, is important in cryptographic hash functions. Previous research has shown that a QUBO can be designed to simulate SHA-1, facilitating the execution of a pre-image attack. In this paper, we present a novel QUBO model with fewer bits, linear terms, and quadratic terms than previous QUBO models for simulating hash values and pre-image calculations in SHA-1. While the previous QUBO model requires 34,016 bits, 32,584 linear terms, and 168,188 quadratic terms, our new model requires only 32,096 bits, 26,600 linear terms, and 157,308 quadratic terms. This approach can efficiently execute pre-image attacks using a quantum annealer, providing valuable insights for evaluating the robustness of cryptographic security systems.