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International Journal of Parallel Emergent and Distributed Systems

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

X. H. Li, Koji Nakano, Shunsuke Tsukiyama, Yasuaki Ito, Takumi Kato, Yuya Kawamata

Published Jul 9, 2024• 2 citations
Publisher

Abstract

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.

DOI
10.1080/17445760.2024.2376928

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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 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.
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 broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ~42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.
The Quadratic Unconstrained Binary Optimization problem (QUBO) has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this article, we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits. We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 79–97 2017
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.
Algorithms and hardware for solving quadratic unconstrained binary optimization (QUBO) problems have made significant recent progress. This advancement has focused attention on formulating combinatorial optimization problems as quadratic polynomials. To improve the performance of solving large QUBO problems, it is essential to minimize the number of binary variables used in the objective function. In this paper, we propose a QUBO formulation that offers a bit capacity advantage over conventional quadratization techniques. As a key application, this formulation significantly reduces the number of binary variables required for score-based Bayesian network structure learning. Experimental results on 16 instances, ranging from 37 to 223 variables, demonstrate that our approach requires fewer binary variables than quadratization by orders of magnitude. Moreover, an annealing machine that implement our formulation have outperformed existing algorithms in score maximization.