Algorithms For Combinatorial Optimization Problems Research Articles

Quantum-Informed Recursive Optimization Algorithms

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component (e.g., locality) and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We showcase the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow circuits of the quantum approximate optimization algorithm, solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a template for a broader class of hybrid quantum-classical algorithms for combinatorial optimization. Published by the American Physical Society 2024

Quantum-Inspired Genetic Algorithms for Combinatorial Optimization Problems

Quantum-Inspired Genetic Algorithms (QIGAs) are a trailblazing force in the ever-evolving field of optimization, combining traditional genetic algorithms with quantum concepts to solve challenging combinatorial problems. By contrasting QIGAs with traditional Genetic Algorithms (GAs) in the setting of the Traveling Salesman Problem (TSP), this study explores the potential of QIGAs. The research reveals the transformational potential of quantum-inspired techniques through a thorough investigation of convergence speed, solution quality, and scalability.

An Ising Model-Based Annealing Processor With 1024 Fully Connected Spins for Combinatorial Optimization Problems

This brief presents a novel annealing processor (AP) design with 1024 fully-connected spins based on a modified Ising model annealing algorithm for combinatorial optimization problems. By adopting the proposed Turbo code-based interleaved random sequence generator (TCSG) and multi-spin update method, the memory usage is made considerable reduction and multi-spin parallel update is supported. The prototype is implemented using FPGA with the operation frequency of 100 MHz. We tested our design on various G-set problems with an average cut accuracy of 99.19% achieved. The proposed design outperforms the CPU-based method by achieving a max speedup of 1099×.

An introduction to variational quantum algorithms for combinatorial optimization problems

An introduction to variational quantum algorithms for combinatorial optimization problems

Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis‐Hastings Warm‐Starting Algorithm

AbstractQuantum approximate optimization algorithm (QAOA) is a promising variational quantum algorithm for combinatorial optimization problems. However, the implementation of QAOA is limited due to the requirement that the problems be mapped to Ising Hamiltonians and the nonconvex optimization landscapes. Although the Ising Hamiltonians for many NP hard problems have been obtained, a general method to obtain the Ising Hamiltonians for constrained combinatorial optimization problems (CCOPs) has not yet been investigated. In this paper, a general method is introduced to obtain the Ising Hamiltonians for CCOPs and the Metropolis‐Hastings warm‐starting algorithm for QAOA is presented which can provably converge to the global optimal solutions. The effectiveness of this method is demonstrated by tackling the minimum weight vertex cover (MWVC) problem, the minimum vertex cover (MVC) problem, and the maximal independent set problem as examples. The Ising Hamiltonian for the MWVC problem is obtained first time by using this method. The advantages of the Metropolis‐Hastings warm‐starting algorithm presented here is numerically analyzed through solving 30 randomly generated MVC cases with 1‐depth QAOA.

Ant-Inspired Metaheuristic Algorithms for Combinatorial Optimization Problems in Water Resources Management

Ant-inspired metaheuristic algorithms known as ant colony optimization (ACO) offer an approach that has the ability to solve complex problems in both discrete and continuous domains. ACOs have gained significant attention in the field of water resources management, since many problems in this domain are non-linear, complex, challenging and also demand reliable solutions. The aim of this study is to critically review the applications of ACO algorithms specifically in the field of hydrology and hydrogeology, which include areas such as reservoir operations, water distribution systems, coastal aquifer management, long-term groundwater monitoring, hydraulic parameter estimation, and urban drainage and storm network design. Research articles, peer-reviewed journal papers and conference papers on ACO were critically analyzed to identify the arguments and research findings to delineate the scope for future research and to identify the drawbacks of ACO. Implementation of ACO variants is also discussed, as hybrid and modified ACO techniques prove to be more efficient over traditional ACO algorithms. These algorithms facilitate formulation of near-optimal solutions, and they also help improve cost efficiency. Although many studies are attempting to overcome the difficulties faced in the application of ACO, some parts of the mathematical analysis remain unsolved. It is also observed that despite its popularity, studies have not been successful in incorporating the uncertainty in ACOs and the problems of dimensionality, convergence and stability are yet to be resolved. Nevertheless, ACO is a potential area for further research as the studies on the applications of these techniques are few.

Multi-Objective Neural Evolutionary Algorithm for Combinatorial Optimization Problems.

There has been a recent surge of success in optimizing deep reinforcement learning (DRL) models with neural evolutionary algorithms. This type of method is inspired by biological evolution and uses different genetic operations to evolve neural networks. Previous neural evolutionary algorithms mainly focused on single-objective optimization problems (SOPs). In this article, we present an end-to-end multi-objective neural evolutionary algorithm based on decomposition and dominance (MONEADD) for combinatorial optimization problems. The proposed MONEADD is an end-to-end algorithm that utilizes genetic operations and rewards signals to evolve neural networks for different combinatorial optimization problems without further engineering. To accelerate convergence, a set of nondominated neural networks is maintained based on the notion of dominance and decomposition in each generation. In inference time, the trained model can be directly utilized to solve similar problems efficiently, while the conventional heuristic methods need to learn from scratch for every given test problem. To further enhance the model performance in inference time, three multi-objective search strategies are introduced in this work. Our experimental results clearly show that the proposed MONEADD has a competitive and robust performance on a bi-objective of the classic travel salesman problem (TSP), as well as Knapsack problem up to 200 instances. We also empirically show that the designed MONEADD has good scalability when distributed on multiple graphics processing units (GPUs).

A hybrid discrete state transition algorithm for combinatorial optimization problems

The discrete state transition algorithm (DSTA) has been wildly applied to deal with combinatorial optimization problems. However, its low convergence accuracy limits its application in large-scale optimization problems. Aiming at the convergence performance and search intensity of the algorithm, a hybrid discrete state transition algorithm (HDSTA) is proposed in this work by introducing tabu search and elite solution set. Firstly, a searching mechanism with the integration of DSTA and tabu search (TS) is established, which allows moving to adjacent solutions at an increased cost to escape from the local optimum. Specifically, a tabu list as adaptive memory is adopted to avoid the loop when deviating from local optima. Secondly, an elite solution set is introduced to integrate the information of the previous optimal solution and the global optimal solution, and the search strategy is modified to expand the range and diversity of candidate solutions. Finally, the proposed HDSTA is verified according to the real data on two well-known optimization problems (staff assignment problem and traveling salesman problem) and the real data of an industrial case. The experimental results show the effectiveness of the proposed algorithm in large-scale optimization problems.

Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach

The impressive progress in quantum hardware in the last years has raised the interest of the quantum computing community in harvesting the computational power of such devices. However, in the absence of error correction, these devices can only reliably implement very shallow circuits or comparatively deeper circuits at the expense of a nontrivial density of errors. In this work, we obtain extremely tight limitation bounds for standard NISQ proposals in both the noisy and noiseless regimes, with or without error-mitigation tools. The bounds limit the performance of both circuit model algorithms, such as QAOA, and also continuous-time algorithms, such as quantum annealing. In the noisy regime with local depolarizing noise $p$, we prove that at depths $L=\mathcal{O}(p^{-1})$ it is exponentially unlikely that the outcome of a noisy quantum circuit outperforms efficient classical algorithms for combinatorial optimization problems like Max-Cut. Although previous results already showed that classical algorithms outperform noisy quantum circuits at constant depth, these results only held for the expectation value of the output. Our results are based on newly developed quantum entropic and concentration inequalities, which constitute a homogeneous toolkit of theoretical methods from the quantum theory of optimal mass transport whose potential usefulness goes beyond the study of variational quantum algorithms.

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [ 31 ] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut , for many others, e.g., Max-SAT and Max-DiCut , the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut , Max-2SAT , and Max-DiCut , and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers p. While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of n spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within (1−ϵ) times the ground state energy. We hope to match its performance with the QAOA.Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the 2p QAOA parameters, in the infinite size limit that can be evaluated on a computer with O(16p) complexity. We evaluate the formula up to p=12, and find that the QAOA at p=11 outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as n→∞, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.

An estimation of distribution algorithm for combinatorial optimization problems

This paper considers solving more than one combinatorial problem considered some of the most difficult to solve in the combinatorial optimization field, such as the job shop scheduling problem (JSSP), the vehicle routing problem with time windows (VRPTW), and the quay crane scheduling problem (QCSP). A hybrid metaheuristic algorithm that integrates the Mallows model and the Moth-flame algorithm solves these problems. Through an exponential function, the Mallows model emulates the solution space distribution for the problems; meanwhile, the Moth-flame algorithm is in charge of determining how to produce the offspring by a geometric function that helps identify the new solutions. The proposed metaheuristic, called HEDAMMF (Hybrid Estimation of Distribution Algorithm with Mallows model and Moth-Flame algorithm), improves the performance of recent algorithms. Although knowing the algebra of permutations is required to understand the proposed metaheuristic, utilizing the HEDAMMF is justified because certain problems are fixed differently under different circumstances. These problems do not share the same objective function (fitness) and/or the same constraints. Therefore, it is not possible to use a single model problem. The aforementioned approach is able to outperform recent algorithms under different metrics for these three combinatorial problems. Finally, it is possible to conclude that the hybrid metaheuristics have a better performance, or equal in effectiveness than recent algorithms.

Hybrid Bird Mating Optimizer With Single-Based Algorithms for Combinatorial Optimization Problems

Bird mating optimizer (BMO) is a population-based metaheuristic that has been recently extended to solve combinatorial optimization problems. Even though the algorithm shows promising performance in solving combinatorial optimization problems, it suffers from slow convergence and poor efficiency which leads to poor solution quality for some problem instances. This is due to the limited capability of BMO in exploiting the search space and identifying more promising regions. Therefore, in this work we propose a hybrid BMO with five single-based metaheuristics: hill-climbing, late acceptance hill-climbing, simulated annealing, iterated greedy heuristic and variable iterated greedy heuristic. Each of these algorithms is used inside the BMO to exploit the search space, and improve the quality of solution generated from the BMO population. This work also compares which one of these five is better for hybridizing with BMO. The performance of these algorithms is tested on two combinatorial problems: travelling salesman problem and berth allocation problem. Experimental results demonstrate that the hybrid algorithm is superior to BMO when applied to both problems and it improved the BMO by 1.13% for BAP and by 4.13% for TSP. Furthermore, the hybrid algorithm is able to match the best-known results for most of the instances. In addition, the proposed hybrid approaches perform well over both tested domains and obtain competitive results when compared to the best-known results that have previously been presented in the scientific literature.

Evolutionary Many-Objective Algorithms for Combinatorial Optimization Problems: A Comparative Study

Many optimization problems encountered in the real-world have more than two objectives. To address such optimization problems, a number of evolutionary many-objective optimization algorithms were developed recently. In this paper, we tested 18 evolutionary many-objective algorithms against well-known combinatorial optimization problems, including knapsack problem (MOKP), traveling salesman problem (MOTSP), and quadratic assignment problem (mQAP), all up to 10 objectives. Results show that some of the dominance and reference-based algorithms such as non-dominated sort genetic algorithm (NSGA-III), strength Pareto-based evolutionary algorithm based on reference direction (SPEA/R), and Grid-based evolutionary algorithm (GrEA) are promising algorithms to tackle MOKP and MOTSP with 5 and 10 while increasing the number of objectives. Also, the dominance-based algorithms such as MaOEA-DDFC as well as the indicator-based algorithms such as HypE are promising to solve mQAP with 5 and 10 objectives. In contrast, decomposition based algorithms present the best on almost problems at saving time. For example, t-DEA displayed superior performance on MOTSP for up to 10 objectives.

Multi-center variable-scale search algorithm for combinatorial optimization problems with the multimodal property

Combinatorial optimization problems (COPs) are discrete problems arising from aerospace, bioinformatics, manufacturing, and other fields. One of the classic COPs is the scheduling problem. Moreover, these problems are usually multimodal optimization problems with a quantity of global and local optima. As a result, many search algorithms can easily become trapped into local optima. In this article, we propose a multi-center variable-scale search algorithm for solving both single-objective and multi-objective COPs. The algorithm consists of two distinct points. First, the multi-center strategy chooses several individuals with better performance as the only parents of the next generation, which means that there are a number of separate searching areas around the searching center. Second, the next generation of the population is produced by a variable-scale strategy with an exponential equation based on the searching center. The equation is designed to control the neighborhood scale, and adaptively realize the large-scale and small-scale searches at different search stages to balance the maintenance of diversity and convergence speed. In addition, an approach of adjusting centers is proposed concerning the number and distribution of centers for solving multi-objective COPs. Finally, the proposed algorithm is applied to three COPs, including the well-known flexible job shop scheduling problem, the unrelated parallel machine scheduling problem, and the test task scheduling problem. Both the single-objective optimization algorithm and the multi-objective optimization algorithm demonstrate competitive performance compared with existing methods.

A novel multi-parent order crossover in genetic algorithm for combinatorial optimization problems

Many multi-parent crossovers have been proposed to solve specific combinatorial optimization problem and not applicable to solve other problems (i.e. cannot produce feasible solution). Only multi-parent partially mapped crossover (MPPMX) and adjacency-based crossover (ABC) have been proposed to work over different combinatorial problems. However, both MPPMX and ABC suffered from a very high computational time or poor performance. Therefore, this work proposes a novel multi-parent order crossover (MPOX) for solving several combinatorial optimization problems with reasonable amount of time. The MPOX extends the two-parent order crossover by modifying the recombination operator to recombine more than two parents and generates a new offspring. MPOX at first selects the crossover points and divides the parents into n substrings based on these points (where n is the number of parents). Then, MPOX copies a predefined number of elements from each parent into the offspring based on their order while checking the feasibility of the offspring. The performance of MPOX is tested on the traveling salesman problems and berth allocation problems, which are widely studied in the literature. Experimental results demonstrated that the MPOX significantly improves the OX in both problem domains and outperforms both ABC and MPPMX over the travelling salesman problem and the berth allocation problem with less computational time. These results indicate the effectiveness of MPOX over OX, ABC and MPPMX, and its capability for solving both problems.

Deterministic Parallel Algorithms for Fooling Polylogarithmic Juntas and the Lovász Local Lemma

Many randomized algorithms can be derandomized efficiently using either the method of conditional expectations or probability spaces with low (almost-) independence. A series of papers, beginning with Luby (1993) and continuing with Berger and Rompel (1991) and Chari et al. (2000), showed that these techniques can be combined to give deterministic parallel algorithms for combinatorial optimization problems involving sums of w -juntas. We improve these algorithms through derandomized variable partitioning, reducing the processor complexity to essentially independent of w and time complexity to linear in w . As a key subroutine, we give a new algorithm to generate a probability space which can fool a given set of neighborhoods. Schulman (1992) gave an NC algorithm to do so for neighborhoods of size w ≤ O (log n ). Our new algorithm is in NC 1 , with essentially optimal time and processor complexity, when w = O (log n ); it remains in NC up to w = polylog( n ). This answers an open problem of Schulman. One major application of these algorithms is an NC algorithm for the Lovász Local Lemma. Previous NC algorithms, including the seminal algorithm of Moser and Tardos (2010) and the work of Chandrasekaran et. al (2013), required that (essentially) the bad-events could span only O (log n ) variables; we relax this to polylog( n ) variables. We use this for an NC 2 algorithm for defective vertex coloring, which works for arbitrary degree graphs.

A Discrete Teaching Learning Based Optimization Algorithm for Combinatorial Optimization Problems

The Teaching Learning Based Optimization algorithm is a population-based algorithm that was first designed for solving continuous optimization problems. A Discrete Teaching Learning Based Optimization algorithm is performed for combinatorial optimization problems in this paper. The proposed Discrete Teaching Learning Based Optimization algorithm uses a path-relinking strategy combined with a local search instead of the ways employed in the original Teaching Learning Based Optimization to move solutions within the search space. Moreover the exploration and exploitation stages were ensured by a Mean solution. The proposed algorithm was tested on 35 benchmarks of the cell formation problem. The results were satisfactory enough to classify the Discrete Teaching Learning Based Optimization algorithm among promising nature-inspired algorithms.

Balanced-evolution genetic algorithm for combinatorial optimization problems: the general outline and implementation of balanced-evolution strategy based on linear diversity index

How to rationally inject randomness to control population diversity is still a difficult problem in evolutionary algorithms. We propose balanced-evolution genetic algorithm (BEGA) as a case study of this problem. Similarity guide matrix (SGM) is a two-dimensional matrix to express the population (or subpopulation) distribution in coding space. Different from binary-coding similarity indexes, SGM is able to be suitable for binary-coding and symbol-coding problems, simultaneously. In BEGA, opposite-direction and forward-direction regions are defined by using two SGMs as reference points, respectively. In opposite-direction region, diversity subpopulation always tries to increase Hamming distances between themselves and the current population. In forward-direction region, intensification subpopulation always tries to decrease Hamming distances between themselves and the current elitism population. Thus, diversity subpopulation is more suitable for injecting randomness. Linear diversity index (LDI) measures the individual density around the center-point individual in coding space, which is characterized by itself linearity. According to LDI, we control the search-region ranges of diversity and intensification subpopulations by using negative and positive perturbations, respectively. Thus, the search efforts between exploration and exploitation are balanced. We compared BEGA with CHC, dual-population genetic algorithm, variable dissortative mating genetic algorithm, quantum-inspired evolutionary algorithm, and greedy genetic algorithm for 12 benchmarks. Experimental results were acceptable. In addition, it is worth noting that BEGA is able to directly solve bounded knapsack problem (i.e. symbol-coding problem) as one EA-based solver, and does not transform bounded knapsack problem into an equivalent binary knapsack problem.

Machine Learning Approaches to Learning Heuristics for Combinatorial Optimization Problems

This paper examines various ways of utilizing machine learning to enhance or develop heuristic and metaheuristic algorithms for combinatorial optimization problems (COPs). A general agent-based framework is introduced, which enables learning both constructive and perturbative heuristics for a wide range of COPs. The framework consists of an agent-based generalized solution algorithm (GSA) to COPs, along with a machine learning architecture and training procedure that would inject intelligence via learning to the agents. A GSA in our framework is an assembly of k agents working in parallel, generating the solution to a given problem one step at a time, either by improving a known solution or by constructing one from scratch. The generated partial solution is then implemented in a simulated or real system that represents the environment, with state changes being fed back to the k agents. In the proposed framework, intelligence can be injected into the agents individually or in groups, via learning from the correlation of their solution decisions with a final objective value. A discussion is also provided on what machine learning approaches to use depending on the availability of training data, which in turn depends on the choice of algorithm architecture.