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arXiv:1911.10680

The deep learning and statistical physics applications to the problems of combinatorial optimization

Semyon Sinchenko, Bazhanov, Dmitry

Published Nov 16, 2019• 2 citations
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Abstract

We present herein a new approach based on the simultaneous application of the deep learning and statistical physics methods to solve the combinatorial optimization problems. The recent modern advanced techniques, such as an artificial neural network, demonstrate their efficiency for solving various physical tasks for the quantum many-body systems, which may be related directly to the problems of combinatorial optimization. One of them is a classical Maximum Cut (MaxCut) problem, which we ascribe here to the search of the ground state of a quantum many-body system using an artificial neural network and deep learning. We found that the exact solution received for a quantum system corresponds to its counterpart in a classical MaxCut problem. As a proof, we have realized our approach for two random graphs of different size, containing 60 vertices (885 edges) and 100 vertices (2475 edges), and achieved for them the total performances p60~0.99 and $p100~0.97 of known maximal cuts, respectively.

DOI
10.48550/arxiv.1911.10680

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We present herein a new approach based on the simultaneous application of the deep learning and statistical physics methods to solve the combinatorial optimization problems. The recent modern advanced techniques, such as an artificial neural network, demonstrate their efficiency for solving various physical tasks for the quantum many-body systems, which may be related directly to the problems of combinatorial optimization. One of them is a classical Maximum Cut (MaxCut) problem, which we ascribe here to the search of the ground state of a quantum many-body system using an artificial neural network and deep learning. We found that the exact solution received for a quantum system corresponds to its counterpart in a classical MaxCut problem. As a proof, we have realized our approach for two random graphs of different size, containing 60 vertices (885 edges) and 100 vertices (2475 edges), and achieved for them the total performances p600.99 and $p1000.97 of known maximal cuts, respectively.
The classical simulation of quantum systems typically requires exponential resources. Recently, the introduction of a machine learning-based wavefunction ansatz has led to the ability to solve the quantum many-body problem in regimes that had previously been intractable for existing exact numerical methods. Here, we demonstrate the utility of the variational representation of quantum states based on artificial neural networks for performing quantum optimization. We show empirically that this methodology achieves high approximation ratio solutions with polynomial classical computing resources for a range of instances of the Maximum Cut (MaxCut) problem whose solutions have been encoded into the ground state of quantum many-body systems up to and including 256 qubits.
Recent advances in deep learning techniques pose a question of whether they can facilitate the task of finding good quality solutions to combinatorial optimization (CO) problems in a practically relevant solution time. Specifically, it is of practical relevance to determine to which extent graph neural networks (GNNs) can be applied to CO problems that can be formulated as QUBO’s and thus be naturally interpreted as graph problems. In this research a GNNsolver is applied to two classical CO problems–the maximum cut problem and maximum independent set problem–in an unsupervised learning setting. We show that while GNN solver consistently finds good quality solutions for the Max Cut problem irrespective of the size and density of the graph, solving MIS problems is challenging for all but very sparse graphs. We further show how this problem can be addressed by embedding transfer between these two problems and compare two different GNN architectures–GCN and GraphSAGE on their robustness with respect to graph density and symmetry. Finally we demonstrate that change of widely used Adam optimizer to Rprop optimizer can lead to considerable reduction of solution times. • Previously suggested GNNsolver is effective on sparse graphs only. • Replacing Adam with RPROP reduced solution times. • GraphSAGE can handle denser graphs, GCN performs better on sparse graphs. • Transfer learning between MaxCut and MIS problems improves solvers performance.
Combinatorial optimization is essential across numerous disciplines. Traditional metaheuristics excel at exploring complex solution spaces efficiently, yet they often struggle with scalability. Deep learning has become a viable alternative for quickly generating high-quality solutions, particularly when metaheuristics underperform. In recent years, quantum-inspired approaches such as tensor networks have shown promise in addressing these challenges. Despite these advancements, a thorough comparison of the different paradigms is missing. This study evaluates eight algorithms on Weighted Max-Cut graphs ranging from 10 to 250 nodes. Specifically, we compare a Genetic Algorithm representing metaheuristics, a Graph Neural Network for deep learning, and the Density Matrix Renormalization Group as a tensor network approach. Our analysis focuses on solution quality and computational efficiency (i.e., time and memory usage). Numerical results show that the Genetic Algorithm achieves near-optimal results for small graphs, although its computation time grows significantly with problem size. The Graph Neural Network offers a balanced solution for medium-sized instances with low memory demands and rapid inference, yet it exhibits more significant variability on larger graphs. Meanwhile, the Tensor Network approach consistently yields high approximation ratios and efficient execution on larger graphs, albeit with increased memory consumption.
In recent years, there has been notable interest in investigating combinatorial optimization (CO) problems by neural-based framework. An emerging strategy to tackle these challenging problems involves the adoption of graph neural networks (GNNs) as an alternative to traditional algorithms, a subject that has attracted considerable attention. Despite the growing popularity of GNNs and traditional algorithm solvers in the realm of CO, there is limited research on their integrated use and the correlation between them within an end-to-end framework. The primary focus of our work is to formulate a more efficient and precise framework for CO by employing decision-focused learning on graphs. Additionally, we introduce a decision-focused framework that utilizes GNNs to address CO problems with auxiliary support. To realize an end-to-end approach, we have designed two cascaded modules: (a) an unsupervised trained graph predictive model, and (b) a solver for quadratic binary unconstrained optimization. Empirical evaluations are conducted on various classical tasks, including maximum cut, maximum independent set, and minimum vertex cover. The experimental results on classical CO problems (i.e. MaxCut, MIS, and MVC) demonstrate the superiority of our method over both the standalone GNN approach and classical methods.
In recent years, quantum computing has emerged as a transformative force in the field of combinatorial optimization, offering novel approaches to tackling complex problems that have long challenged classical computational methods. Among these, the Quantum Approximate Optimization Algorithm (QAOA) stands out for its potential to efficiently solve the Max-Cut problem, a quintessential example of combinatorial optimization. However, practical application faces challenges due to current limitations on quantum computational resource. Our work optimizes QAOA initialization, using Graph Neural Networks (GNN) as a warm-start technique. This sacrifices affordable computational resource on classical computer to reduce quantum computational resource overhead, enhancing QAOA's effectiveness. Experiments with various GNN architectures demonstrate the adaptability and stability of our framework, highlighting the synergy between quantum algorithms and machine learning. Our findings show GNN's potential in improving QAOA performance, opening new avenues for hybrid quantum-classical approaches in quantum computing and contributing to practical applications.
In recent years, quantum computing has emerged as a transformative force in the field of combinatorial optimization, offering novel approaches to tackling complex problems that have long challenged classical computational methods. Among these, the Quantum Approximate Optimization Algorithm (QAOA) stands out for its potential to efficiently solve the Max-Cut problem, a quintessential example of combinatorial optimization. However, practical application faces challenges due to current limitations on quantum computational resource. Our work optimizes QAOA initialization, using Graph Neural Networks (GNN) as a warm-start technique. This sacrifices affordable computational resource on classical computer to reduce quantum computational resource overhead, enhancing QAOA's effectiveness. Experiments with various GNN architectures demonstrate the adaptability and stability of our framework, highlighting the synergy between quantum algorithms and machine learning. Our findings show GNN's potential in improving QAOA performance, opening new avenues for hybrid quantum-classical approaches in quantum computing and contributing to practical applications.