Combinatorial Optimization Problems Research Articles

A Q-learning-based algorithm for the block relocation problem

The Block Relocation Problem (BRP), also known as the Container Relocation Problem, is a challenging combinatorial optimization problem in block stacking systems and has many applications in real-world scenarios such as logistics and manufacturing industry. The BRP is about finding the optimal way to retrieve blocks from a storage area with the objective of minimizing the number of relocations. The BRPs have been studied for a long time, and have been solved primarily using conventional optimization techniques, including mathematical programming models, as well as both exact and heuristic algorithms. For the first time, this paper tackles the problem using a reinforcement learning method. We focus on one of the major variants of the BRP—the restricted BRP with duplicate priorities (RBRP-dup). We first model the RBRP-dup as a Markov decision process and then propose a Q-learning-based algorithm to solve the problem. The Q-learning-based algorithm contains two phases. In the learning phase, two innovative mechanisms: an optimal rule-integrated behaviour policy and a heuristic-based dynamic initialization method, are incorporated into the Q-learning model to reduce the size of the state-action space and accelerate convergence. In the optimization phase, the insights obtained in the learning phase are combined with a heuristic algorithm to improve decision-making. The performance of our proposed method is evaluated against the state-of-the-art exact algorithm and a commonly used heuristic algorithm based on benchmark instances from the literature. The computational experiments demonstrate the superiority of our proposed method regarding solution quality in large and complex instances.

Comparing three generations of D-wave quantum annealers for minor embedded combinatorial optimization problems

Comparing three generations of D-wave quantum annealers for minor embedded combinatorial optimization problems

Algorithm for determining the operating parameters of an electrical network in the problem of optimal reconfiguration in real time

Relevance. The need to develop technical solutions to reduce electrical energy losses in medium-voltage distribution electrical networks of oil and gas fields. Aim. Development of an algorithm for determining the parameters of the operating mode of an electrical network, applicable in a system of optimal control of the network configuration in order to reduce the level of electricity losses in the network. Methods. Mathematical modeling methods and combinatorial optimization methods. The nodal voltage method and the Newton–Raphson method are used to calculate the network operating mode. The mode is corrected using a sensitivity matrix. Results and conclusions. The authors have developed an algorithm for determining the parameters of the electric network operating mode when determining its optimal configuration. Optimization of the optimal configuration is performed based on the criterion of minimum active power losses in lines, taking into account the limitations on the power reserve of the feeding transformer substations and the number of normal breaks on one line. The developed algorithm is based on solving nonlinear systems of equations of nodal voltages using the Newton–Raphson method, as well as on correcting the parameters of the previously calculated operating mode using the sensitivity matrix. The optimal network configuration is determined by solving the combinatorial optimization problem using the branch and bound method. The operation of the algorithm is illustrated by the example of a network section containing two overhead power lines with two power sources and three feeding transformer substations. Based on the results of the analysis of all possible network configurations determined by combinations of normal breaks, it is concluded that the use of the algorithm will reduce the level of active power losses by up to 23.8%, which will increase the efficiency of the electric grid complex.

Ant Colony Optimization (ACO) for Traveling Salesman Problem: A Review

The traveling salesman problem (TSP) is a fundamental combinatorial optimization problem with applications in resource management, logistics, and communications. In order to address TSP and its differences, this paper discusses developments in Ant Colony Optimization (ACO), a biologically inspired algorithm. Inspired by the foraging activity of ants, ACO's decentralized and recursive methodology has proven successful in solving difficult routing problems. ACO's scalability, convergence speed, and solution quality have been greatly enhanced over time through innovations including hybridization with algorithms such as Firefly, genetic algorithms, parallel computing frameworks, and adaptation mechanisms. These developments have given the ACO the flexibility and efficiency to handle dynamic situations, such as real-time vehicle guidance and underwater navigation. Despite its progress, issues remain such as scalability in resource-limited contexts, processing overhead, and reliance on parameter modification. This work summarizes current developments in ACO, noting how revolutionary the TSP solution is, pointing out its drawbacks, and suggesting areas for further study. Leveraging emerging technologies like machine learning and quantum computing, ACO has huge potential to progressively address challenging real-world problems. This review provides a comprehensive framework for developing uses of ACOs and reaffirms their status as a key component of improvement research.

Automated design of state transition rules in ant colony optimization by genetic programming: a comprehensive investigation

The automated design of Ant Colony Optimization (ACO) algorithms has become increasingly significant, particularly in addressing complex combinatorial optimization problems. Although existing methods have achieved some success, they still face limitations, particularly the high dependency on expert knowledge, pre-solved data, and challenges in interpretability. Genetic Programming (GP), as a proven technology, has shown potential in optimizing the automated design state transition rules of ACO. However, existing research on GP-ACO is insufficient, particularly in terms of experimental validation and systematic evaluation. To address these issues, this study conducts comprehensive experiments to explore several key questions: the generality of GP-ACO on homogeneously distributed maps, the impact of different ACO variants on the learning capabilities of GP-ACO, the effect of 2-opt local search on the learning capabilities of GP-ACO, the enhancement of learning capabilities through the addition of more global information, and the interpretability of GP-ACO. The findings indicate that GP-ACO exhibits robust generality; variations among ACO variants have minimal impact on learning performance; 2-opt local search can somewhat diminish the performance of GP-ACO in the Max–Min Ant System; additional global information can significantly enhance the learning capabilities of GP-ACO; and GP-ACO has good interpretability.

An Improved Marriage in Honey-Bee Optimization Algorithm for Minimizing Earliness/Tardiness Penalties in Single-Machine Scheduling with a Restrictive Common Due Date

This study evaluates the efficiency of a swarm intelligence algorithm called marriage in honey-bee optimization (MBO) in solving the single-machine weighted earliness/tardiness problem, a type of NP-hard combinatorial optimization problem. The goal is to find the optimal sequence for completing a set of tasks on a single machine, minimizing the total penalty incurred for tasks being completed too early or too late compared to their deadlines. To achieve this goal, the study adapts the MBO metaheuristic by introducing modifications to optimize the objective function and produce high-quality solutions within reasonable execution times. The novelty of this work lies in the application of MBO to the single-machine weighted earliness/tardiness problem, an approach previously unexplored in this context. MBO was evaluated using the test problem set from Biskup and Feldmann. It achieved an average improvement of 1.03% across 280 problems, surpassing upper bounds in 141 cases (50.35%) and matching or exceeding them in 193 cases (68.93%). In the most constrained problems (h = 0.2 and h = 0.4), the method achieved an average improvement of 3.77%, while for h = 0.6 and h = 0.8, the average error was 1.72%. Compared to other metaheuristics, MBO demonstrated competitiveness, with a maximum error of 1.12%. Overall, MBO exhibited strong competitiveness, delivering significant improvements and high efficiency in the problems studied.

A Systematic Review on Reinforcement Learning for Industrial Combinatorial Optimization Problems

This paper presents a systematic review on reinforcement learning approaches for combinatorial optimization problems based on real-world industrial applications. While this topic is increasing in popularity, explicit implementation details are not always available in the literature. The main objective of this paper is characterizing the agent–environment interactions, namely, the state space representation, action space mapping and reward design. Also, the main limitations for practical implementation and the needed future developments are identified. The literature selected covers a wide range of industrial combinatorial optimization problems, found in the IEEE Xplore, Scopus and Web of Science databases. A total of 715 unique papers were extracted from the query. Then, out-of-scope applications, reviews, surveys and papers with insufficient implementation details were removed. This resulted in a total of 298 papers that align with the focus of the review with sufficient implementation details. The state space representation shows the most variety, while the reward design is based on combinations of different modules. The presented studies use a large variety of features and strategies. However, one of the main limitations is that even with state-of-the-art complex models the scalability issues of increasing problem complexity cannot be fully solved. No methods were used to assess risk of biases or automatically synthesize the results.

Systematic Literature Review of Optimization Algorithms for P||Cmax Problem

In the era of open data and open science, it is important that, before announcing their new results, authors consider all previous studies and ensure that they have competitive material worth publishing. To save time, it is popular to replace the exhaustive search of online databases with the utilization of generative Artificial Intelligence (AI). However, especially for problems in niche domains, generative AI results may not be precise enough and sometimes can even be misleading. A typical example is P||Cmax, an important scheduling problem studied mainly in a wider context of parallel machine scheduling. As there is an uncovered symmetry between P||Cmax and other similar optimization problems, it is not easy for generative AI tools to include all relevant results into search. Therefore, to provide the necessary background data to support researchers and generative AI learning, we critically discuss comparisons between algorithms for P||Cmax that have been presented in the literature. Thus, we summarize and categorize the “state-of-the-art” methods, benchmark test instances, and compare methodologies, all over a long time period. We aim to establish a framework for fair performance evaluation of algorithms for P||Cmax, and according to the presented systematic literature review, we uncovered that it does not exist. We believe that this framework could be of wider importance, as the identified principles apply to a plethora of combinatorial optimization problems.

Missing value replacement in strings and applications

Missing values arise routinely in real-world sequential (string) datasets due to: (1) imprecise data measurements; (2) flexible sequence modeling, such as binding profiles of molecular sequences; or (3) the existence of confidential information in a dataset which has been deleted deliberately for privacy protection. In order to analyze such datasets, it is often important to replace each missing value, with one or more valid letters, in an efficient and effective way. Here we formalize this task as a combinatorial optimization problem: the set of constraints includes the context of the missing value (i.e., its vicinity) as well as a finite set of user-defined forbidden patterns, modeling, for instance, implausible or confidential patterns; and the objective function seeks to minimize the number of new letters we introduce. Algorithmically, our problem translates to finding shortest paths in special graphs that contain forbidden edges representing the forbidden patterns. Our work makes the following contributions: (1) we design a linear-time algorithm to solve this problem for strings over constant-sized alphabets; (2) we show how our algorithm can be effortlessly applied to fully sanitize a private string in the presence of a set of fixed-length forbidden patterns [Bernardini et al. 2021a]; (3) we propose a methodology for sanitizing and clustering a collection of private strings that utilizes our algorithm and an effective and efficiently computable distance measure; and (4) we present extensive experimental results showing that our methodology can efficiently sanitize a collection of private strings while preserving clustering quality, outperforming the state of the art and baselines. To arrive at our theoretical results, we employ techniques from formal languages and combinatorial pattern matching.

Stochastic logic in biased coupled photonic probabilistic bits

Optical computing often employs tailor-made hardware to implement specific algorithms, trading generality for improved performance in key aspects like speed and power efficiency. An important computing approach that is still missing its corresponding optical hardware is probabilistic computing, used e.g. for solving difficult combinatorial optimization problems. In this study, we propose an experimentally viable photonic approach to solve arbitrary probabilistic computing problems. Our method relies on the insight that coherent Ising machines composed of coupled and biased optical parametric oscillators can emulate stochastic logic. We demonstrate the feasibility of our approach by using numerical simulations equivalent to the full density matrix formulation of coupled optical parametric oscillators.

Resource‐Efficient Adaptive Variational Quantum Algorithm for Combinatorial Optimization Problems

AbstractVariational quantum algorithms (VQAs) are quantum‐classical hybrid algorithms that are promising for the near future. The Quantum Approximate Optimization Algorithm (QAOA) is a representative VQA for solving combinatorial optimization problems. However, the parameterized quantum circuit (PQC) of QAOA still has room for improvement. The existing method, called ADAPT‐QAOA, has improved the PQC of QAOA, but the circuit depth remains deep. A Resource‐Efficient Adaptive VQA (RE‐ADAPT‐VQA) that utilizes gates from a new gate pool is proposed to construct a PQC. RE‐ADAPT‐VQA incrementally integrates parameterized quantum gates based on the gradient until the predefined stopping criteria are satisfied, and a rollback mechanism is proposed to ensure that the circuit remains shallow. The algorithm is experimentally simulated to solve Max‐Cut problem and the maximum independent set problem. The results show that RE‐ADAPT‐VQA significantly reduces circuit depth, single‐qubit gates, and CNOT gates compared to existing methods, while maintaining the same level of energy error.

Solution Algorithms for the Capacitated Location Tree Problem with Interconnections

This paper addresses the Capacitated Location Tree Problem with Interconnections, a new combinatorial optimization problem with applications in network design. In this problem, the required facilities picked from a set of potential facilities must be opened to serve customers using a tree-shaped network. Costs and capacities are associated with the opening of facilities and the establishment of network links. Customers have a given demand that must be satisfied while respecting the facilities and link capacities. The problem aims to minimize the total cost of designing a distribution network while considering facility opening costs, demand satisfaction, capacity constraints, and the creation of interconnections to enhance network resilience. A valid mixed-integer programming was proposed and an exact solution method based on the formulation was used to solve small- and medium-sized instances. To solve larger instances two metaheuristic approaches were used. A specific decoder procedure for the metaheuristic solution approaches was also proposed and used to help find solutions, especially for large instances. Computational experiments and results using the three solution approaches are also presented. Finally, a case study on the design of electrical transportation systems was presented and solved.

Efficient Frontier and Applications in Product Offering and Pricing

Problem definition: The joint assortment and pricing problem is notoriously difficult to solve, especially for large-scale issues subject to various operational constraints arising from business practices. In this paper, we delve into the joint optimization challenge under constrained logit choice models, accounting for product-differentiated price sensitivities across static, randomized, and dynamic contexts. Methodology/results: We develop a unified optimization methodology that applies efficient frontier and dimensional reduction and transforms the joint optimization problem into a single-variable one with respect to the total choice probability. The efficient frontier is employed to transform the nonconcave objective function equivalently into its concave counterpart. We show that the optimal prices are uniquely determined by the common target adjusted markup. The complexity of the mixed combinatorial optimization problem can be reduced by searching over efficient sets of polynomial size. In the randomized assortment and pricing problem in which the product offer sets and prices follow certain distribution, we show that randomization has the potential to elevate the total revenue to the exact efficient frontier, a feat that the static problem may fail to achieve (with a given total choice probability). For dynamic joint product selection and pricing, we find that the optimal policy adopts a simple time-threshold structure that can be precomputed. Furthermore, we illustrate the robustness of our methodology by extending the analysis to a variety of general settings, including the nested logit model. Managerial implications: The proposed methodology contributes to the increasingly popular topic in retail management by significantly reducing the computational complexity of joint product offering and pricing under different constraints. Our results regarding the effects of product set constraints and price bounds provide valuable insights and guidance to practitioners on product offering and pricing in various business scenarios. Funding: C. Ke acknowledges financial support from the National Natural Science Foundation of China [Grant 72101113]. L. Lu acknowledges financial support from the Hong Kong Research Grants Council [Grants 26501021 and 16502423]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0164 .

A bi-subpopulation coevolutionary immune algorithm for multi-objective combinatorial optimization in multi-UAV task allocation

With the development of Unmanned Aerial Vehicle (UAV) technology towards multi-UAV and UAV swarm, multi-UAV cooperative task allocation has more and more influence on the success or failure of UAV missions. From the operational research point of view, such problems belong to high-dimensional combinatorial optimization problems, which makes the solving process face many challenges. One is that the discrete and high-dimensional decision variables make the quality of the solution obtained with acceptable time not guaranteed. Second, the desired solution of real missions often needs to satisfy multiple objective functions, or a set of solutions for decision-making. Therefore, this paper constructs a Multi-objective Combinatorial Optimization in Multi-UAV Task Allocation Problem (MCOTAP) model, and proposes a Bi-subpopulation Coevolutionary Immune Algorithm (BCIA). The two coevolutionary mechanisms improve the lower limit of population diversity, and the evolutionary strategy pool integrating multiple strategies and the adaptive strategy selection mechanism enhance the local search ability in the late evolution. In the experiments, BCIA competes fairly with the mainstream multi-objective evolutionary algorithms (MOEAs), multi-objective immune algorithms (MOIAs) and the recently proposed multi-UAV mission planning algorithms. The experimental results on different test problems (including several multi-objective combinatorial optimization benchmark problems and the proposed MCOTAP model) show that BCIA has superior performance in solving multi-objective combinatorial optimization problems (MCOPs). At the same time, the effectiveness of each design component of BCIA has been comprehensively verified in the ablation study.

Recursive Quantum Relaxation for Combinatorial Optimization Problems

Quantum optimization methods use a continuous degree-of-freedom of quantum states to heuristically solve combinatorial problems, such as the MAX-CUT problem, which can be attributed to various NP-hard combinatorial problems. This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. The code is available at https://github.com/ToyotaCRDL/rqrao.

Efficient digital quadratic unconstrained binary optimization solvers for SAT problems

Abstract Boolean satisfiability (SAT) is a propositional logic problem of determining whether an assignment of variables satisfies a Boolean formula. Many combinatorial optimization problems can be formulated in Boolean SAT logic -- either as k-SAT decision problems or Max k-SAT optimization problems, with conflict-driven (CDCL) solvers being the most prominent. Despite their ability to handle large instances, CDCL-based solvers have fundamental scalability limitations. In light of this, we propose recently-developed quadratic unconstrained binary optimization (QUBO) solvers as an alternative platform for 3-SAT problems. To utilize them, we implement a 2-step [3-SAT]-[Max 2-SAT]-[QUBO] conversion procedure and present a rigorous proof to explicitly calculate the number of both satisfied and violated clauses of the original 3-SAT instance from the transformed Max 2-SAT formulation. We then demonstrate, through numerical simulations on several benchmark instances, that digital QUBO solvers can achieve state-of-the-art accuracy on 78-variable 3-SAT benchmark problems. Our work facilitates the broader use of quantum annealers on noisy intermediate-scale quantum (NISQ) devices, as well as their quantum-inspired digital counterparts, for solving 3-SAT problems.

Study of Meta-Learning Attempts to Select Algorithm for Hard Combinatorial Optimisation Problems

Study of Meta-Learning Attempts to Select Algorithm for Hard Combinatorial Optimisation Problems

Enhancing computational accuracy with parallel parameter optimization in variational quantum eigensolver

Variational quantum algorithms have promising applications in noisy intermediate-scale quantum (NISQ) devices. These algorithms rely on a classical optimization outer loop that minimizes a parameterized quantum circuit function. The optimization in variational quantum eigensolver (VQE) is NP-hard, meaning that finding the optimal solution is infeasible in the worst-case scenario. One way to address this challenge is through parallel optimization of parameters using multiple-parameterized quantum circuits. However, this approach is unsuitable for cloud-based quantum processing unit utilization due to the increased number of quantum circuit executions. Although NISQ devices have limitations in terms of gate depth, their size has been growing in recent years. Therefore, implementing multiple-parameterized quantum circuits in NISQ devices can suppress the increase in the number of executions. In this study, we propose a parallel-VQE, which leverages the parallel execution of parameterized quantum circuits to perform parallel parameter optimization in VQE, achieving convergence to solutions closer to the ground state. We validate the effectiveness of parallel-VQE in solving the random weighted max-cut problem using numerical simulations and a real quantum device. We present the results of running up to six circuits in parallel (120 qubits) and demonstrate the advantages of using multiple units to improve computational accuracy. This study provides a potential method for solving eigenvalue problems and combinatorial optimization problems for future quantum devices.

Test Case Minimization with Quantum Annealers

Quantum annealers are specialized quantum computers for solving combinatorial optimization problems with special quantum computing characteristics, e.g., superposition and entanglement. Theoretically, quantum annealers can outperform classic computers. However, current quantum annealers are constrained by a limited number of qubits and cannot demonstrate quantum advantages. Nonetheless, research is needed to develop novel mechanisms to formulate combinatorial optimization problems for quantum annealing (QA). However, QA applications in software engineering remain unexplored. Thus, we propose BootQA , the very first effort at solving test case minimization (TCM) problems on classical software with QA. We provide a novel TCM formulation for QA and utilize bootstrap sampling to optimize the qubit usage. We also implemented our TCM formulation in three other optimization processes: simulated annealing (SA), QA without problem decomposition, and QA with an existing D-Wave problem decomposition strategy, and conducted an empirical evaluation with three real-world TCM datasets. Results show that BootQA outperforms QA without problem decomposition and QA with the existing decomposition strategy regarding effectiveness. Moreover, BootQA ’s effectiveness is similar to SA. Finally, BootQA has higher efficiency in terms of time when solving large TCM problems than the other three optimization processes.

Searching for oil storage tank layout by solving NP-complete combinatorial optimisation problem

In the process of designing a new tank farm or reconstructing an old one, we often face the task of finding the optimal layout of hydrocarbon storage tanks. It is necessary to comply with both the customer's wishes and the requirements of normative documents regarding restrictions in the location of petroleum product storage facilities, and it must be correlated with the size of the site where the tanks are to be constructed. This results in an optimisation problem that can be solved in various ways. In this research, we shall consider two methods for solving such an NP-complete combinatorial optimisation problem: the knapsack problem and the bin-packing problem. For this purpose, we shall solve the discussed problem using both methods analytically, and then form programme algorithms for the solution based on the Python language. Let us compare both methods in real cases of tank placement and analyse their efficiency.