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Frontiers in Computer Science

Partitioning QUBO with two-way one-hot conditions on traveling salesman problems for city distributions with multiple clusters

Akihiro Yatabe

Published Apr 26, 2024• 2 citations
Publisher

Abstract

We introduce a method for solving a quadratic unconstrained binary optimization (QUBO) with the two-way one-hot constraints by dividing the QUBO into parts and solving it with an Ising machine. The one-hot constraint is a constraint condition in that only one binary variable takes the value one for a set of multiple binary variables. The two-way one-hot constraint imposes one-hot constraint conditions on every row and every column of a two-dimensional array of binary variables with equal numbers of rows and columns. For example, traveling salesman problems (TSPs) have two-way one-hot constraints. We propose two methods to solve a TSP by dividing the cities into clusters based on information in the QUBO matrix. The proposed methods decompose cities into clusters and solve TSPs for each cluster. We solve TSPs such that the cities are distributed in clusters and compare the results of the two proposed methods with the results of solving the problem without partitioning. The results show that the proposed methods are robust to the coefficient parameters in the Hamiltonian of QUBO and can obtain a solution closer to the optimal solution when solving the problem without partitioning is hard. We also discuss the application of the methods proposed in the TSPs to the quadratic assignment problems and to the problems of ordering things.

DOI
10.3389/fcomp.2024.1285244

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We introduce a method for solving a quadratic unconstrained binary optimization (QUBO) with the two-way one-hot constraints by dividing the QUBO into parts and solving it with an Ising machine. The one-hot constraint is a constraint condition in that only one binary variable takes the value one for a set of multiple binary variables. The two-way one-hot constraint imposes one-hot constraint conditions on every row and every column of a two-dimensional array of binary variables with equal numbers of rows and columns. For example, traveling salesman problems (TSPs) have two-way one-hot constraints. We propose two methods to solve a TSP by dividing the cities into clusters based on information in the QUBO matrix. The proposed methods decompose cities into clusters and solve TSPs for each cluster. We solve TSPs such that the cities are distributed in clusters and compare the results of the two proposed methods with the results of solving the problem without partitioning. The results show that the proposed methods are robust to the coefficient parameters in the Hamiltonian of QUBO and can obtain a solution closer to the optimal solution when solving the problem without partitioning is hard. We also discuss the application of the methods proposed in the TSPs to the quadratic assignment problems and to the problems of ordering things.
Ising machines have attracted much attention in recent years because of their potential in solving combinatorial optimization problems that are difficult to be solved using conventional computers. Ising machines solve the optimization problem that can be transformed into a quadratic unconstrained binary optimization (QUBO) problem. In the case of a problem with constraints, named a constraint problem, an objective function and constraint function with penalty coefficients are combined into a single Hamiltonian of a QUBO problem. When solving a constraint problem, the accuracy of the solution decreases compared to a QUBO problem with an objective function only. This is because penalty coefficients are set to excessively large values to prevent constraint violations. As a result, it becomes difficult to minimize the objective function. To solve this problem, this paper proposes a method that partitions constraint functions and reduces the value of the penalty coefficients. The performance evaluation using the traveling salesperson problem (TSP) with one-hot constraints demonstrates that the proposed method can improve the solution accuracy compared to the conventional method.
Recently, a clustering method using a combinatorial optimization problem, called combinatorial clustering, has been drawing attention due to the rapid spreads of quantum annealing and simulated annealing.Combinatorial clustering is performed by minimizing an objective function under a condition to satisfy a one-hot constraint.The objective function and the constraint function are generally formulated to a unified objective function of a QUBO (Quadratic Unconstrained Binary Optimization) problem using the method of the Lagrange multiplier.The coefficients of the QUBO function can be represented by a square matrix, which is called the QUBO matrix.Although the Lagrange multiplier needs to be large enough to avoid violating the constraint, it is usually hard to be set appropriately due to the limitation of the bit precision.For example, the latest quantum annealer can handle values represented by only six or fewer bits.Even conventional computing systems cannot control the larger value of the Lagrange multiplier as the number of data points increases.Besides, the execution time for combinatorial clustering increases exponentially as the problem size increases.This is because the time for the QUBO matrix generation is long and a dominant factor of the total execution time when the problem size is large.To solve these problems, this paper proposes combinatorial clustering that overcomes the limitation of the method of the Lagrange multiplier.The proposed method uses a QUBO solver that can externally define the one-hot constraint independent from the objective function, and the externally-defined constraint is satisfied by the operations with multiple bit-flips.As the QUBO function contains only the objective function, the method of the Lagrange multiplier is not necessary.The proposed method can optimize the objective function sufficiently even when the problem size is large.Since the constraint function is not included in the QUBO function, the proposed method also reduces the time for the QUBO matrix generation.The experimental results obtained using the artificial and real data show that the proposed method can improve the quality of annealing-based clustering results in all data sets.The experimental results also clarify that the quality of the proposed method is almost equal to or better than that of quasi-optimal clustering methods such as K-means.The evaluation using the multiple traveling salesman problem shows that the proposed method can obtain shorter tour lengths than the conventional annealing-based clustering in all 13 cases and K-means++ in 6 out of 12 cases with a significant difference.Furthermore, the proposed method can accelerate the execution time for combinatorial clustering because there is no need to calculate the coefficients of the constraint function.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
We present a map from the travelling salesman problem (TSP), a prototypical NP-complete combinatorial optimisation task, to the ground state associated with a system of many-qudits. Conventionally, the TSP is cast into a quadratic unconstrained binary optimisation (QUBO) problem, that can be solved on an Ising machine. The size of the corresponding physical system's Hilbert space is 2N22^{N^2}2N2, where NNN is the number of cities considered in the TSP. Our proposal provides a many-qudit system with a Hilbert space of dimension 2Nlog⁡2N2^{N\log_2N}2Nlog2​N, which is considerably smaller than the dimension of the Hilbert space of the system resulting from the usual QUBO map. This reduction can yield a significant speedup in quantum and classical computers. We simulate and validate our proposal using variational Monte Carlo with a neural quantum state, solving the TSP in a linear layout for up to almost 100 cities.
Ising machines are specialized solvers for combinatorial optimization problems (COPs), which are typically formulated as quadratic unconstrained binary optimization (QUBO) models. To enhance their performance, various methods have been proposed to construct subQUBOs by reducing the number of variables in the original QUBO. This study proposes a general subQUBO construction method applicable to a wide range of problems. The variables are clustered based on their interaction structure using theK-means clustering. A subQUBO is constructed from a randomly selected cluster and a pre-obtained tentative solution and then solved. This process of cluster selection and subQUBOs is performed iteratively. We evaluated the performance of the method on the quadratic assignment problem and compared it with two baselines: one that randomly reduces variables and another that directly solves the original QUBO without any variable reduction. Experimental evaluations demonstrated that the proposed method achieves high-accuracy solutions within the same computation time, particularly for large problem sizes.
Recently, a clustering method using a combinatorial optimization problem, called combinatorial clustering, has been drawing attention due to the rapid spreads of quantum annealing and simulated annealing. Combinatorial clustering is performed by minimizing an objective function under a condition to satisfy a constraint. The objective function and the constraint function are generally formulated to a unified objective function of a QUBO (Quadratic Unconstrained Binary Optimization) problem by the method of the Lagrange multiplier.Although the Lagrange multiplier needs to be large enough to avoid violating the one-hot constraint, the Lagrange multiplier usually cannot be set appropriately due to the limitation of the bit precision. The latest quantum annealer can handle values represented by only six or fewer bits. Even conventional computing systems cannot handle the larger value of the Lagrange multiplier as the number of data points increases.To solve this problem, this paper proposes combinatorial clustering that overcomes the limitations of the method of the Lagrange multiplier. The proposed method uses a QUBO solver that can externally define the one-hot constraint independent from the objective function, and the externally-defined constraint is satisfied by the bit-flip operations. Since the constraint function is not included in the objective function, it is no longer needed to determine the Lagrange multiplier. The experimental results show that the proposed method can improve the quality of clustering results.