Solution Of Combinatorial Optimization Problems Research Articles

Frequency tunable CMOS ring oscillator‐based Ising machine

SummaryOscillator‐based Ising machines (OIMs) particularly those realized in complementary metal oxide semiconductor (CMOS) have gained popularity for solving combinatorial optimization problems (COPs) in recent years due to its scalability, low‐power consumption, and room temperature operation. The implemented OIMs have thus far focused on solving optimization problems with a single global minima. However, real‐life optimization problems often have multiple solutions. In this paper, we propose a generalized approach to solve COPs with single (without contention), as well as multiple (with contention) solutions using frequency tunable CMOS ring oscillator (ROSC)‐based Ising machine. A capacitive frequency tunable CMOS ring‐oscillator coupled with an internal subharmonic injection locking (SHIL) generator realized using 14‐nm FinFET models works as Ising spin in the proposed approach. We demonstrate how frequency tuning can help in attaining good quality results and also determine all possible solutions of COP with contention. We also propose a generalized algorithm for monitoring the states of the oscillator network to indicate tuning necessity and extract solutions from the oscillator's output irrespective of the type of COP.

The effect of classical optimizers and Ansatz depth on QAOA performance in noisy devices

The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm for Near-term Intermediate-Scale Quantum computers (NISQ) providing approximate solutions for combinatorial optimization problems. The QAOA utilizes a quantum-classical loop, consisting of a quantum ansatz and a classical optimizer, to minimize some cost function, computed on the quantum device. This paper presents an investigation into the impact of realistic noise on the classical optimizer and the determination of optimal circuit depth for the Quantum Approximate Optimization Algorithm (QAOA) in the presence of noise. We find that, while there is no significant difference in the performance of classical optimizers in a state vector simulation, the Adam and AMSGrad optimizers perform best in the presence of shot noise. Under the conditions of real noise, the SPSA optimizer, along with ADAM and AMSGrad, emerge as the top performers. The study also reveals that the quality of solutions to some 5 qubit minimum vertex cover problems increases for up to around six layers in the QAOA circuit, after which it begins to decline. This analysis shows that increasing the number of layers in the QAOA in an attempt to increase accuracy may not work well in a noisy device.

Genetic Engineering Algorithm (GEA): An Efficient Metaheuristic Algorithm for Solving Combinatorial Optimization Problems

Genetic Algorithms (GAs) are known for their efficiency in solving combinatorial optimization problems, thanks to their ability to explore diverse solution spaces, handle various representations, exploit parallelism, preserve good solutions, adapt to changing dynamics, handle combinatorial diversity, and provide heuristic search. However, limitations such as premature convergence, lack of problem-specific knowledge, and randomness of crossover and mutation operators make GAs generally inefficient in finding an optimal solution. To address these limitations, this paper proposes a new metaheuristic algorithm called the Genetic Engineering Algorithm (GEA) that draws inspiration from genetic engineering concepts. GEA redesigns the traditional GA while incorporating new search methods to isolate, purify, insert, and express new genes based on existing ones, leading to the emergence of desired traits and the production of specific chromosomes based on the selected genes. Comparative evaluations against stateof-the-art algorithms on benchmark instances demonstrate the superior performance of GEA, showcasing its potential as an innovative and efficient solution for combinatorial optimization problems.

Stochastic Simulated Quantum Annealing for Fast Solution of Combinatorial Optimization Problems

Stochastic Simulated Quantum Annealing for Fast Solution of Combinatorial Optimization Problems

Quafu-Qcover: Explore combinatorial optimization problems on cloud-based quantum computers

We introduce Quafu-Qcover, an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends. Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm (QAOA). It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization (QUBO) model and its corresponding Ising model, which can be subsequently transformed into a weight graph. The core of Qcover relies on a graph decomposition-based classical algorithm, which efficiently derives the optimal parameters for the shallow QAOA circuit. Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers. Compared to a general-purpose compiler, our compiler demonstrates the ability to generate shorter circuit depths, while also exhibiting superior speed performance. Additionally, the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time, utilizing the most recent calibration data from the superconducting quantum devices. This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity. The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time, enabling asynchronous processing. Moreover, it incorporates modules for results preprocessing and visualization, facilitating an intuitive display of solutions for combinatorial optimization problems. We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.

Quantum approximate optimization algorithm parameter prediction using a convolutional neural network

The Quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm aiming to produce approximate solutions for combinatorial optimization problems. In the QAOA, the quantum part prepares a quantum parameterized state that encodes the solution, where the parameters are optimized by a classical optimizer. However, it is difficult to find optimal parameters when the quantum circuit becomes deeper. Hence, there is numerous active research on the performance and the optimization cost of QAOA. In this work, we build a convolutional neural network to predict parameters of depth p + 1 QAOA instance by the parameters from the depth p QAOA counterpart. We propose two strategies based on this model. First, we recurrently apply the model to generate a set of initial values for a certain depth QAOA. It successfully initiates depth 10 QAOA instances, whereas each model is only trained with the parameters from depths less than 6. Second, the model is applied repetitively until the maximum expected value is reached. An average approximation ratio of 0.9759 for Max-Cut over 264 Erdős–Rényi graphs is obtained, while the optimizer is only adopted for generating the first input of the model.

Adaptive solution prediction for combinatorial optimization

This paper aims to predict optimal solutions for combinatorial optimization problems (COPs) via machine learning (ML). To find high-quality solutions efficiently, existing work uses a ML prediction of the optimal solution to guide heuristic search, where the ML model is trained offline under the supervision of solved problem instances with known optimal solutions. To predict the optimal solution with sufficient accuracy, it is critical to provide a ML model with adequate features that can effectively characterize decision variables. However, acquiring such features is challenging due to the high complexity of COPs. This paper proposes a framework that can better characterize decision variables by harnessing feedback from a heuristic search over several iterative steps, enabling an offline-trained ML model to predict the optimal solution in an adaptive manner. We refer to this approach as adaptive solution prediction (ASP). Specifically, we employ a set of statistical measures as features, which can extract useful information from feasible solutions found by a heuristic search and inform the ML model as to which value a decision variable is likely to take in high-quality solutions. Our experiments on three NP-hard COPs show that ASP substantially improves the prediction quality of an offline-trained ML model and achieves competitive results compared to several heuristic methods in terms of solution quality. Furthermore, we demonstrate that ASP can be used as a heuristic-pricing method for column generation, to boost an exact branch-and-price algorithm for solving the graph coloring problem.

Initial State Encoding via Reverse Quantum Annealing and H-Gain Features.

Quantum annealing is a specialized type of quantum computation that aims to use quantum fluctuations in order to obtain global minimum solutions of combinatorial optimization problems. Programmable D-Wave quantum annealers are available as cloud computing resources, which allow users low-level access to quantum annealing control features. In this article, we are interested in improving the quality of the solutions returned by a quantum annealer by encoding an initial state into the annealing process. We explore twoD-Wave features that allow one toencode such an initialstate: the reverse annealing (RA) and theh-gain(HG)features.RAaimstorefineaknownsolutionfollowinganannealpathstartingwithaclassical state representing a good solution, going backward to a point where a transverse field is present, and then finishing the annealing process with a forward anneal. The HG feature allows one to put a time-dependent weighting scheme on linear () biases of the Hamiltonian, and we demonstrate that this feature likewise can be used to bias the annealing to start from an initial state. We also consider a hybrid method consisting of a backward phase resembling RA and a forward phase using the HG initial state encoding. Importantly, we investigate the idea of iteratively applying RA and HG to a problem, with the goal of monotonically improving on an initial state that is not optimal. The HG encoding technique is evaluated on a variety of input problems including the edge-weighted maximum cut problem and the vertex-weighted maximum clique problem, demonstrating that the HG technique is a viable alternative to RA for some problems. We also investigate how the iterative procedures perform for both RA and HG initial state encodings on random whole-chip spin glasses with the native hardware connectivity of the D-Wave Chimera and Pegasus chips.

Neuro-Ising: Accelerating Large-Scale Traveling Salesman Problems via Graph Neural Network Guided Localized Ising Solvers

One of the most extensively studied combinatorial optimization problems is the Travelling Salesman Problem (TSP). Considerable research efforts in the past have resulted in exact solvers. However, the runtime of such hand-crafted solutions increases exponentially with problem size. Ising model based solvers have also gained prominence due to their abilities to find fast and approximate solutions for combinatorial optimization problems. However, such Ising based heuristics also suffer from scalability as the solution quality becomes increasingly sub-optimal with increase in problem size. In this work, we propose Neuro-Ising – a machine learning framework which uses Ising models to find clusters of near-optimal partial solutions of large scale TSPs and combines those solutions by employing a supervised data driven mechanism, which we model as a Graph Neural Network (GNN). The GNN is trained from solution instances obtained through exact solvers and hence, the proposed approach generalizes to unseen problems while avoiding the run-time complexity otherwise required, if the solution is built from scratch. Using standard computing resources, our proposed framework rapidly converges to near-optimal solutions for 15 TSPs (upto <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 5k$ </tex-math></inline-formula> cities) from the TSPLib benchmark suite. We report <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10.66\times $ </tex-math></inline-formula> speedup over Tabu Search for 8 problems. Furthermore, compared to two state-of-the-art clustering-based TSP solvers, Neuro-Ising achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 38 \times $ </tex-math></inline-formula> faster convergence along with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 8.9\%$ </tex-math></inline-formula> better quality of solution, on average.

Constraint Preserving Mixers for the Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most of the literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints. We generalize the “XY”-mixer designed to preserve the subspace of “one-hot” states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of the number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for an “XY”-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.

Quantum Optimization via Four-Body Rydberg Gates.

A large ongoing research effort focuses on obtaining a quantum advantage in the solution of combinatorial optimization problems on near-term quantum devices. A particularly promising platform implementing quantum optimization algorithms are arrays of trapped neutral atoms, laser coupled to highly excited Rydberg states. However, encoding combinatorial optimization problems in atomic arrays is challenging due to limited interqubit connectivity of the native finite-range interactions. Here, we present a four-body Rydberg parity gate, enabling a direct and straightforward implementation of the parity architecture, a scalable architecture for encoding arbitrarily connected interaction graphs. Our gate relies on adiabatic laser pulses and is fully programmable by adjusting two hold times during operation. We numerically demonstrate implementations of the quantum approximate optimization algorithm (QAOA) for small-scale test problems. Variational optimization steps can be implemented with a constant number of system manipulations, paving the way for experimental investigations of QAOA beyond the reach of numerical simulations.

A distributed adaptive optimization spiking neural P system for approximately solving combinatorial optimization problems

An optimization spiking neural P system (OSNPS) aims to obtain the approximate solutions of combinatorial optimization problems without the aid of evolutionary operators of evolutionary algorithms or swarm intelligence algorithms. To develop the promising and significant research direction, this paper proposes a distributed adaptive optimization spiking neural P system (DAOSNPS) with a distributed population structure and a new adaptive learning rate considering population diversity. Extensive experiments on knapsack problems show that DAOSNPS gains much better solutions than OSNPS, adaptive optimization spiking neural P system, genetic quantum algorithm and novel quantum evolutionary algorithm. Population diversity and convergence analysis indicate that DAOSNPS achieves a better balance between exploration and exploitation than OSNPS and AOSNPS.

Spintronics-compatible Approach to Solving Maximum-Satisfiability Problems with Probabilistic Computing, Invertible Logic, and Parallel Tempering

The search of hardware-compatible strategies for solving NP-hard combinatorial optimization problems (COPs) is an important challenge of today s computing research because of their wide range of applications in real world optimization problems. Here, we introduce an unconventional scalable approach to face maximum satisfiability problems (Max-SAT) which combines probabilistic computing with p-bits, parallel tempering, and the concept of invertible logic gates. We theoretically show the spintronic implementation of this approach based on a coupled set of Landau-Lifshitz-Gilbert equations, showing a potential path for energy efficient and very fast (p-bits exhibiting ns time scale switching) architecture for the solution of COPs. The algorithm is benchmarked with hard Max-SAT instances from the 2016 Max-SAT competition (e.g., HG-4SAT-V150-C1350-1.cnf which can be described with 2851 p-bits), including weighted Max-SAT and Max-Cut problems.

Quantum Analog Annealing of Gain‐Dissipative Ising Machine Driven by Colored Gaussian Noise

AbstractGain‐dissipative Ising machines (GIMs) are a type of quantum analog equipment that can rapidly determine the optimal solution for combinatorial optimization problems. When the noise intensity is significantly lower than the fixed point of the system, the performance of a GIM is not influenced by the fluctuation of the noise intensity. However, the noise in this study is limited to Gaussian white noise. The influence of prevalent colored noise on GIMs has not been researched. In this study, the influence of common‐colored noise on the performance of GIMs is numerically investigated. The results of a domain clustering dynamics analysis reveal that red noise can better suppress the generation of the noise‐induced irregular temporary domain. Furthermore, several prevalent MAXCUT problem topologies, including the Moebius ladder, random Moebius ladder, and 2D random lattice, are adopted as test benchmarks. The results reveal that GIMs influenced by white, blue, and violet noise perform better at low‐intensity noise condition. In contrast, pink and red noise‐injected GIMs demonstrate higher performances when applied to MAXCUT topologies with both ferromagnetic and antiferromagnetic connections under larger noise intensity conditions. This indicates that the noise dispersion can be used as an additional hyperparameter to optimize the performance of GIMs.

Flowing Water Algorithm: A New Approach for Combinatorial Optimization Problems

Being greatly inspired by the natural flowing regulation of water, we propose a new meta-heuristic algorithm — <i>Flowing Water Algorithm</i> (FWA) for the solution of combinatorial optimization problems (COPs). Since the solution space of COPs is multidimensional, complex and has many local extreme values, according to our proposed method, it appears to be similar to an endless hilly area with mountains, valleys and plateaus. The downward-flowing water in such area finds its way to the lowest point in the hill. Water always flows downward and eventually converges at the lowest place without any outside intervention except for gravity. Such a flowing course can be deemed as a process for the water to seek for the lowest point. The proposed algorithm is derived from such a water flow process. This algorithm combines a local search strategy with a population-based search strategy to improve both local and global search abilities. Four operators, including the local search, water overflow, drilling water tunnel and evaporation-rain are included in FWA, making this algorithm successfully perform tabu search, positive feedback, “survival of the fittest”, and local optimum escape. Two examples of its application in the traveling salesman problem (TSP) show that FWA outperforms the benchmark methods for both solution quality and convergence speed.

Deep reinforcement learning with credit assignment for combinatorial optimization

Recent advances in Deep Reinforcement Learning (DRL) demonstrates the potential for solving Combinatorial Optimization (CO) problems. DRL shows advantages over traditional methods both on scalability and computation efficiency. However, the DRL problems transformed from CO problems usually have a huge state space, and the main challenge of solving them has changed from high computation complexity to high sample complexity.Credit assignment determines the contribution of each internal decision to the final success or failure, and it has been shown to be effective in reducing the sample complexity of the training process. In this paper, we resort to a model-based reinforcement learning method to assign credits for model-free DRL methods. Since heuristic methods plays an important role on state-of-the-art solutions for CO problems, we propose using a model to represent those heuristic knowledge and derive the credit assignment from the model. This model-based credit assignment can facilitate the model-free DRL to perform a more effective exploration, and the data collected by the model-free DRL refines the model continuously as the training progresses. Extensive experiments on various CO problems with different settings show that our framework outperforms previous state-of-the-art methods on performance and training efficiency.

Quantitative Evaluation of Hardware Binary Stochastic Neurons

Recently there has been increasing activity to build dedicated Ising Machines to accelerate the solution of combinatorial optimization problems by expressing these problems as a ground-state search of the Ising model. A common theme of such Ising Machines is to tailor the physics of underlying hardware to the mathematics of the Ising model to improve some aspect of performance that is measured in speed to solution, energy consumption per solution or area footprint of the adopted hardware. One such approach to build an Ising spin, or a binary stochastic neuron (BSN), is a compact mixed-signal unit based on a low-barrier nanomagnet based design that uses a single magnetic tunnel junction (MTJ) and three transistors (3T-1MTJ) where the MTJ functions as a stochastic resistor (1SR). Such a compact unit can drastically reduce the area footprint of BSNs while promising massive scalability by leveraging the existing Magnetic RAM (MRAM) technology that has integrated 1T-1MTJ cells in ~Gbit densities. The 3T-1SR design however can be realized using different materials or devices that provide naturally fluctuating resistances. Extending previous work, we evaluate hardware BSNs from this general perspective by classifying necessary and sufficient conditions to design a fast and energy-efficient BSN that can be used in scaled Ising Machine implementations. We connect our device analysis to systems-level metrics by emphasizing hardware-independent figures-of-merit such as flips per second and dissipated energy per random bit that can be used to classify any Ising Machine.

PERFORMANCE ANALYSIS OF OPTIMIZATION METHODS FOR SOLVING TRAVELING SALESMAN PROBLEM

The subject of this research is distance and time of several city tour problems which known as traveling salesman problem (tsp). The goal is to find out the gaps of distance and time between two types of optimization methods in traveling salesman problem: exact and approximate. Exact method yields optimal solution but spends more time when the number of cities is increasing and approximate method yields near optimal solution even optimal but spends less time than exact methods. The task in this study is to identify and formulate each algorithm for each method, then to run each algorithm with the same input and to get the research output: total distance, and the last to compare both methods: advantage and limitation. Methods used are Brute Force (BF) and Branch and Bound (B&amp;B) algorithms which are categorized as exact methods are compared with Artificial Bee Colony (ABC), Tabu Search (TS) and Simulated Annealing (SA) algorithms which are categorized as approximate methods or known as a heuristics method. These three approximate methods are chosen because they are effective algorithms, easy to implement and provide good solutions for combinatorial optimization problems. Exact and approximate algorithms are tested in several sizes of city tour problems: 6, 9, 10, 16, 17, 25, 42, and 58 cities. 17, 42 and 58 cities are derived from tsplib: a library of sample instances for tsp; and others are taken from big cities in Java (West, Central, East) island. All of the algorithms are run by MATLAB program. The results show that exact method is better in time performance for problem size less than 25 cities and both exact and approximate methods yield optimal solution. For problem sizes that have more than 25 cities, approximate method – Artificial Bee Colony (ABC) yields better time which is approximately 37% less than exact and deviates 0.0197% for distance from exact method. The conclusion is to apply exact method for problem size that is less than 25 cities and approximate method for problem size that is more than 25 cities. The gap of time will be increasing between two methods when sample size becomes larger.

High-performance combinatorial optimization based on classical mechanics.

Quickly obtaining optimal solutions of combinatorial optimization problems has tremendous value but is extremely difficult. Thus, various kinds of machines specially designed for combinatorial optimization have recently been proposed and developed. Toward the realization of higher-performance machines, here, we propose an algorithm based on classical mechanics, which is obtained by modifying a previously proposed algorithm called simulated bifurcation. Our proposed algorithm allows us to achieve not only high speed by parallel computing but also high solution accuracy for problems with up to one million binary variables. Benchmarking shows that our machine based on the algorithm achieves high performance compared to recently developed machines, including a quantum annealer using a superconducting circuit, a coherent Ising machine using a laser, and digital processors based on various algorithms. Thus, high-performance combinatorial optimization is realized by massively parallel implementations of the proposed algorithm based on classical mechanics.

On the Problem of a Linear Function Localization on Permutations

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.