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Scientific Reports

A tree search algorithm towards solving Ising formulated combinatorial optimization problems

Yunuo Cen, Debasis Das, Xuanyao Fong

Published Aug 30, 2022• 7 citations
Publisher

Abstract

Simulated annealing (SA) attracts more attention among classical heuristic algorithms because many combinatorial optimization problems can be easily recast as the ground state search problem of the Ising Hamiltonian. However, for practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm selects the best near-optimal solution within the feasible search space by exploring along the tree in the sense of "look ahead". Furthermore, motivated by the coherent Ising machine (CIM), the discrete representation of spin states is relaxed to a continuous representation with a regularization term, which enables the use of the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of the proposed algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that with the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated combinatorial optimization problems.

DOI
10.1038/s41598-022-19102-x

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Simulated annealing (SA) attracts more attention among classical heuristic algorithms because many combinatorial optimization problems can be easily recast as the ground state search problem of the Ising Hamiltonian. However, for practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm selects the best near-optimal solution within the feasible search space by exploring along the tree in the sense of "look ahead". Furthermore, motivated by the coherent Ising machine (CIM), the discrete representation of spin states is relaxed to a continuous representation with a regularization term, which enables the use of the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of the proposed algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that with the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated combinatorial optimization problems.
Simulated annealing (SA) attracts more attention among classical heuristic algorithms because the solution of the combinatorial optimization problem can be naturally mapped to the ground state of the Ising Hamiltonian. However, in practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm will select the best near-optimal solution within the feasible search space by exploring along the tree in the sense of `look ahead'. Furthermore, motivated by coherent Ising machine (CIM), we relax the discrete representation of spin states to continuous representation with a regularization term and utilize the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of this algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that above the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated NP-optimization problems.
Abstract Combinatorial optimization problems can be solved by heuristic algorithms such as simulated annealing (SA) which aims to find the optimal solution within a large search space through thermal fluctuations. This algorithm generates new solutions through Markov-chain Monte Carlo techniques which can result in severe limitations, such as slow convergence and a tendency to stay within the same local search space at small temperatures. To overcome these shortcomings, we use the variational classical annealing (VCA) framework that combines autoregressive recurrent neural networks (RNNs) with traditional annealing to sample solutions that are uncorrelated. In this paper, we demonstrate the potential of using VCA as an approach to solving real-world optimization problems. We explore VCA’s performance in comparison with SA at solving three popular optimization problems: the maximum cut problem (Max-Cut), the nurse scheduling problem (NSP), and the traveling salesman problem (TSP). For all three problems, we find that VCA outperforms SA on average in the asymptotic limit by one or more orders of magnitude in terms of relative error. Interestingly, we reach large system sizes of up to 256 cities for the TSP. We also conclude that in the best case scenario, VCA can serve as a great alternative when SA fails to find the optimal solution.
The Ising machine, as a quantum-inspired computing system, can be used to efficiently solve combinatorial optimization problems. Ongoing studies have positioned it to potentially surpass the performance limitations of traditional computers. However, such Ising machines also suffer from scalability as the solution quality becomes sub-optimal when the problem size increases. In this work, we propose an Ising-based framework that uses a distributed Ising algorithm to find the near-optimal solution for large-scale Max-Cut problems through parallel processing via graph segmentation and reassembly. We selected the coherent Ising machine as the hardware solver and successfully found the solution for a 2000-node instance utilizing 20 computational bits on the Gset standard dataset. Additionally, we investigated the key factors influencing the system's performance and discovered that setting the number of reorganizations to one results in higher accuracy. Building on this configuration, we utilized a 100-bit CIM hardware platform (CPQC-1) to solve instances with varying numbers of nodes. The experimental results agree well with the theoretical simulations, confirming the practical feasibility of our method. Moreover, compared to existing Ising-based solvers, our algorithm significantly reduces computational resource requirements while achieving superior performance on large-scale combinatorial optimization problems, highlighting its effectiveness on quantum-inspired hardware.
Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinatorial optimization problems. However, quantum annealing cannot be used to improve a candidate solution obtained by other algorithms because it starts from an initial state where all spin configurations are found with a uniform probability. In this study, we propose an Ising formulation of integer optimization problems to utilize quantum annealing in the iterative improvement strategy. Our formulation exploits the biased sampling of degenerated ground states in transverse magnetic field quantum annealing. We also analytically show that a first-order phase transition is successfully avoided for a fully connected ferromagnetic Potts model if the overlap between a ground state and a candidate solution exceeds a threshold. The proposed formulation is applicable to a wide range of integer optimization problems and enables us to hybridize quantum annealing with other optimization algorithms.
Abstract Combinatorial optimization problems and ground state problems of spin glasses are crucial in various fields of science and technology. However, they often belong to the computational class of NP-hard, presenting significant computational challenges. Traditional algorithms inspired by statistical physics like simulated annealing have been widely adopted. Recently, advancements in Ising machines, such as quantum annealers and coherent Ising machines, offer new paradigms for solving these problems efficiently by embedding them into the analog evolution of nonlinear dynamical systems. However, existing dynamics-based algorithms often suffer from low convergence rates and local minima traps. In this work, we introduce the dual mean-field dynamics into Ising machines. The approach integrates the gradient force and the transverse force into the dynamics of Ising machines in solving combinatorial optimization problems, making it easier for the system to jump out of the local minimums and allowing the dynamics to explore wider in configuration space. We conduct extensive numerical experiments using the Sherrington–Kirkpatrick spin glass up to 10 000 spins and the maximum cut problems with the standard G-set benchmarks. The numerical results demonstrate that our dual mean-field dynamics approach enhances the performance of base Ising machines, providing a more effective solution for large-scale combinatorial optimization problems.
Computers based on physical systems are increasingly anticipated to overcome the impending limitations on digital computer performance. One such computer is a coherent Ising machine (CIM) for solving combinatorial optimization problems. Here, we report a CIM with 100,512 degenerate optical parametric oscillator pulses working as the Ising spins. We show that the CIM delivers fine solutions to maximum cut problems of 100,000-node graphs drastically faster than standard simulated annealing. Moreover, the CIM, when operated near the phase transition point, provides some extremely good solutions and a very broad distribution. This characteristic will be useful for applications that require fast random sampling such as machine learning.