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ACM/IEEE Design Automation Conference (DAC)

Solving traveling salesman problems via a parallel fully connected ising machine

Qichao Tao, Jie Han

Published Jul 10, 2022• 21 citations
Publisher

Abstract

Annealing-based Ising machines have shown promising results in solving combinatorial optimization problems. As a typical class of these problems, however, traveling salesman problems (TSPs) are very challenging to solve due to the constraints imposed on the solution. This article proposes a parallel annealing algorithm for a fully connected Ising machine that significantly improves the accuracy and performance in solving constrained combinatorial optimization problems such as the TSP. Unlike previous parallel annealing algorithms, this improved parallel annealing (IPA) algorithm efficiently solves TSPs using an exponential temperature function with a dynamic offset. Compared with digital annealing (DA) and momentum annealing (MA), the IPA reduces the run time by 44.4 times and 19.9 times for a 14-city TSP, respectively. Large scale TSPs can be more efficiently solved by taking a k-medoids clustering approach that decreases the average travel distance of a 22-city TSP by 51.8% compared with DA and by 42.0% compared with MA. This approach groups neighboring cities into clusters to form a reduced TSP, which is then solved in a hierarchical manner by using the IPA algorithm.

DOI
10.1145/3489517.3530595

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Annealing-based Ising machines have shown promising results in solving combinatorial optimization problems. As a typical class of these problems, however, traveling salesman problems (TSPs) are very challenging to solve due to the constraints imposed on the solution. This article proposes a parallel annealing algorithm for a fully connected Ising machine that significantly improves the accuracy and performance in solving constrained combinatorial optimization problems such as the TSP. Unlike previous parallel annealing algorithms, this improved parallel annealing (IPA) algorithm efficiently solves TSPs using an exponential temperature function with a dynamic offset. Compared with digital annealing (DA) and momentum annealing (MA), the IPA reduces the run time by 44.4 times and 19.9 times for a 14-city TSP, respectively. Large scale TSPs can be more efficiently solved by taking a k-medoids clustering approach that decreases the average travel distance of a 22-city TSP by 51.8% compared with DA and by 42.0% compared with MA. This approach groups neighboring cities into clusters to form a reduced TSP, which is then solved in a hierarchical manner by using the IPA algorithm.
Many combinatorial optimization problems can be solved by numerically simulating classical nonlinear Hamiltonian systems based on the Ising model. Solving the traveling salesman problem (TSP) using the Ising model requires a quadratically increasing number of spins with strict constraints. Unlike classical simulated annealing, simulated bifurcation (SB) can update the states of spins in parallel. This feature can potentially accelerate the convergence of Hamiltonian in the Ising model by taking advantage of modern multi-core processors. As an improved SB algorithm, the ballistic SB (bSB) algorithm is considered for solving the TSP in this paper. The TSP is converted to an Ising problem with external magnetic fields. bSB is then expanded by introducing a time-dependent factor. Experiments on benchmark datasets show that the bSB-based Ising solver offers superior performance in solution quality and convergence speed.
When solving a combinatorial optimization problem with Ising machines, we have to formulate it onto an energy function of Ising model. Here, how to determine the penalty coefficients in the energy function is a great concern if it includes constraint terms. In this paper, we focus on a traveling salesman problem (TSP, in short), one of the combinatorial optimization problems with equality constraints. Firstly, we investigate the relationship between the penalty coefficient and the accuracy of solutions in a TSP. Based on it, we propose a method to obtain a TSP quasi-optimum solution, which is called multiple coefficients trial method. In our proposed method, we use an Ising machine to solve a TSP by changing a penalty coefficient every trial, the TSP solutions can converge very fast in total. Compared to naive methods using simulated-annealing-based Ising machines, we confirmed that our proposed method can reduce the total number of annealing iterations to 1/10 to 1/1000 to obtain a quasi-optimum solution in 32-city TSPs.
One of the vital roles of computing is to solve large-scale combinatorial optimization problems in a short time. In recent years, methods have been proposed that map optimization problems to ones of searching for the ground state of an Ising model by using a stochastic process. Simulated annealing (SA) is a representative algorithm. However, it is inherently difficult to perform a parallel search. Here we propose an algorithm called momentum annealing (MA), which, unlike SA, updates all spins of fully connected Ising models simultaneously and can be implemented on GPUs that are widely used for scientific computing. MA running in parallel on GPUs is 250 times faster than SA running on a modern CPU at solving problems involving 100 000 spin Ising models.
Annealing processors that can quickly solve combinatorial optimization problems (COPs) are attracting attention. They solve COPs by converting them into spin-spin interactions and external fields of the Ising model. Annealing processors that use fully-coupled spin coupling can solve a wide range of COPs. However, it is difficult to execute parallel annealing in fully-coupled spin coupling because one spin is updated by referring to all the other spins. In this paper, we proposed multi-state pseudo annealing (MSPA), which is an annealing method for changing the spin value from a conventional binary to a multi state that takes into account intermediate states. In a software validation, we solved the Maxcut problem and the traveling salesman problem and confirmed that the proposed method improves the solution quality in parallel annealing compared with simulated annealing and pseudo annealing. In addition, we implemented MSPA on a field-programmable gate array as a 1024 -spin 16 -parallel system using a scalable structure. With the proposed system, we solved the Max-cut problem and confirmed that it achieves a 9.40 times faster solving speed and yields a 1% larger Max-cut value compared with the conventional parallelized annealing processors.
Annealing machines use an Ising model to represent combinatorial optimization problems (COPs) and minimize the energy of the model with spin-flip sequences. Pseudo temperature is a key hyperparameter to control the search performance of annealing machines. In general, the temperature is statically scheduled such that it is gradually decreased from a sufficiently high to a sufficiently low values. However, the search process during high and low temperatures in solving constrained COPs does not improve the solution quality as expected, which requires repeated preliminary annealing for pre-tuning. This paper proposes a flip-count-based dynamic temperature control (FDTC) method to make the preliminary annealing unnecessary. FDTC checks whether the current temperature is effective by evaluating the average number of flipped spins in a series of steps. The simulation results for traveling salesman problems and quadratic assignment problems demonstrate that FDTC can obtain comparable or higher solution quality than the static temperature scheduling pre-tuned for every COP.
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.