Combinatorial Optimization Research Articles

How to improve “construct, merge, solve and adapt"? Use reinforcement learning!

AbstractIn this work, we propose a new variant of construct, merge, solve, and adapt (CMSA), which is a recently introduced hybrid metaheuristic for combinatorial optimization. Our newly proposed variant, named reinforcement learning CMSA (RL-CMSA), makes use of a reinforcement learning (RL) mechanism trained online with data gathered during the search process. In addition to generally outperforming standard CMSA, this new variant proves to be more flexible as it does not require a greedy function for the evaluation of solution components at each solution construction step. We present RL-CMSA as a general framework for enhancing CMSA by leveraging a simple RL learning process. Moreover, we study a range of specific designs for the employed learning mechanism. The advantages of the introduced CMSA variant are demonstrated in the context of the far from most string and minimum dominating set problems, showing the improvement in performance and simplicity with respect to standard CMSA. In particular, the best performing RL-CMSA variant proposed is statistically significantly better than the standard algorithm for both problems, obtaining 1.28% and 0.69% better results on average respectively.

Binary Particle Swarm Optimization for Fair User Association in Network Slicing-Enabled Heterogeneous O-RANs

The Open-Radio Access Network (O-RAN) alliance is leading the evolution of telecommunications towards a greater intelligence, openness, virtualization, and interoperability within mobile networks. The O-RAN standard incorporates of many components the Open-Central Unit (O-CU) and Open-Distributed Unit (O-DU), network slicing and heterogeneous base stations (BS). Together, these innovations have given rise to a three-tiered user association (UA) relationship in a type of network called heterogeneous network (HetNet) with network slicing-enabled. There is an absence of efficient UA schemes for achieving fair resource allocation in such network scenario. Hence, this study formulates the fairness-aware UA problem as a utility-based combinatorial optimization problem, which is computationally hard to solve. Hence, an efficient Binary Particle Swarm Optimization (BPSO)-based UA scheme is proposed to solve the problem. Through simulations of an O-RAN based HetNet with network slicing-enabled, performance of the proposed BPSO-UA scheme is compared against two other baseline UA schemes. Results demonstrate the effectiveness of the proposed BPSO-UA scheme in achieving high fairness through equitable network slicing resource allocation, thereby leading to higher user connectivity rate and comparable average spectral efficiency. This innovative approach sheds light on the potential of metaheuristic algorithms in tackling intricate UA challenges, offering valuable insights for the future design and optimization of mobile networks.

An introduction to variational quantum algorithms for combinatorial optimization problems

An introduction to variational quantum algorithms for combinatorial optimization problems

DIRT/µ - Automated extraction of root hair traits using combinatorial optimization.

Similar to any microscopic appendages, such as cilia or antennae, phenotyping of root hairs has been a challenge due to their complex intersecting arrangements in two-dimensional (2D) images and the technical limitations of automated measurements. Digital Imaging of Root Traits at Microscale (DIRT/μ) addresses this issue by computationally resolving intersections and extracting individual root hairs from 2D microscopy images. This solution enables automatic and precise trait measurements of individual root hairs. DIRT/μ rigorously defines a set of rules to resolve intersecting root hairs and minimizes a newly designed cost function to combinatorically identify each root hair in the microscopy image. As a result, DIRT/μ accurately measures traits such as root hair length (RHL) distribution and root hair density (RHD), which are impractical for manual assessment. We tested DIRT/μ on three datasets to validate its performance and showcase potential applications. By measuring root hair traits in a fraction of the time manual methods require, DIRT/μ eliminates subjective biases from manual measurements. Automating individual root hair extraction accelerates phenotyping and quantifies trait variability within and among plants, creating new possibilities to characterize root hair function and their underlying genetics.

A Gnn-Enhanced Ant Colony Optimization for Security Strategy Orchestration

The expansion of Internet of Things (IoT) technology and the rapid increase in data in smart grid business scenarios have led to a need for more dynamic and adaptive security strategies. Traditional static security measures struggle to meet the evolving low-voltage security requirements of state grid systems under this new IoT-driven environment. By incorporating symmetry in metaheuristic algorithms, we can further improve performance and robustness. Symmetrical properties have the potential to lead to more efficient and balanced solutions, improving the overall stability of the grid. We propose a gnn-enhanced ant colony optimization method for orchestrating grid security strategies, which trains across combinatorial optimization problems (COPs) that are representative scenarios in the state grid business scenarios, to learn specific mappings from instances to their heuristic measures. The learned heuristic metrics are embedded into the ant colony optimization (ACO) to generate the optimal security policy adapted to the current security situation. Compared to the ACO and adaptive elite ACO, our method reduces the average time consumption of finding a path within a limited time in the capacitated vehicle routing problem by 67.09% and 66.98%, respectively. Additionally, ablation experiments verify the effectiveness and necessity of the individual functional modules.

A neural algorithm for computing bipartite matchings

Finding optimal bipartite matchings-e.g., matching medical students to hospitals for residency, items to buyers in an auction, or papers to reviewers for peer review-is a fundamental combinatorial optimization problem. We found a distributed algorithm for computing matchings by studying the development of the neuromuscular circuit. The neuromuscular circuit can be viewed as a bipartite graph formed between motor neurons and muscle fibers. In newborn animals, neurons and fibers are densely connected, but after development, each fiber is typically matched (i.e., connected) to exactly one neuron. We cast this synaptic pruning process as a distributed matching (or assignment) algorithm, where motor neurons "compete" with each other to "win" muscle fibers. We show that this algorithm is simple to implement, theoretically sound, and effective in practice when evaluated on real-world bipartite matching problems. Thus, insights from the development of neural circuits can inform the design of algorithms for fundamental computational problems.

Inferring cancer type-specific patterns of metastatic spread.

The metastatic spread of a cancer can be reconstructed from DNA sequencing of primary and metastatic tumours, but doing so requires solving a challenging combinatorial optimization problem. This problem often has multiple solutions that cannot be distinguished based on current maximum parsimony principles alone. Current algorithms use ad hoc criteria to select among these solutions, and decide, a priori, what patterns of metastatic spread are more likely, which is itself a key question posed by studies of metastasis seeking to use these tools. Here we introduce Metient, a freely available open-source tool which proposes multiple possible hypotheses of metastatic spread in a cohort of patients and rescores these hypotheses using independent data on genetic distance of metastasizing clones and organotropism. Metient is more accurate and is up to 50x faster than current state-of-the-art. Given a cohort of patients, Metient can calibrate its parsimony criteria, thereby identifying shared patterns of metastatic dissemination in the cohort. Reanalyzing metastasis in 169 patients based on 490 tumors, Metient automatically identifies cancer type-specific trends of metastatic dissemination in melanoma, high-risk neuroblastoma and non-small cell lung cancer. Metient's reconstructions usually agree with semi-manual expert analysis, however, in many patients, Metient identifies more plausible migration histories than experts, and further finds that polyclonal seeding of metastases is more common than previously reported. By removing the need for hard constraints on what patterns of metastatic spread are most likely, Metient introduces a way to further our understanding of cancer type-specific metastatic spread.

Q-learning based hyper-heuristic with clustering strategy for combinatorial optimization: A case study on permutation flow-shop scheduling problem

In this paper, a Q-learning based hyper-heuristic with clustering strategy (QHH/CS) is proposed for combinatorial optimization problems (COPs). In QHH/CS, a clustering strategy based on low-dimensional mapping method is devised to map initial population to a low-dimensional space, thus obtaining multiple subpopulations accounting for different search directions. To discover more promising search regions around each subpopulation, we propose a parallel Q-learning search mechanism composed of multiple search components, including multi-subpopulation Q-table, state extraction method, contribution-driven reward function, and deep mining local search actions. Relying on these search components, QHH/CS identifies the variations of the objective values of subpopulations to evaluate the solution features of COPs, whereby valuable information can be learned during the search process of the algorithm. To illustrate the effectiveness of QHH/CS, it is applied to solve the permutation flow-shop scheduling problem. We additionally assess QHH/CS through the well-known vehicle routing problem, which confirms the general search ability of the algorithm for COPs. Moreover, the convergence analysis of the QHH/CS algorithm is performed, providing theoretical guidance for the optimization process of the proposed algorithm. Results of experiments demonstrate that QHH/CS can find high-quality solutions to the solved problems.

Hysteretic response to different modes of ramping an external field in sparse and dense Ising spin glasses

We consider the hysteretic behavior of Ising spin glasses at T=0 for various modes of driving. Previous studies mostly focused on an infinitely slow speed Ḣ by which the external field H was ramped to trigger avalanches of spin flips by starting with destabilizing a single spin while few have focused on the effect of different driving methods. First, we show that this conventional protocol imposes a system size dependence. Then, we numerically analyze the response of Ising spin glasses at rates Ḣ that are fixed as well, to elucidate the differences in the response. Specifically, we compare three different modes of ramping (Ḣ=c/N, Ḣ=c/N, and Ḣ=c for constant c) for two types of spin glass systems of size N, representing dense networks by the Sherrington–Kirkpatrick model and sparse networks by the lattice spin glass in d=3 dimensions known as the Edwards Anderson model. Depending on the mode of ramping, we find that the response of each system, in form of spin-flip avalanches and other observables, can vary considerably. In particular, in the N-independent mode applied to the lattice spin glass, which is closest to experimental reality, we observe a percolation transition with a broad avalanche distribution between phases of localized and system-spanning responses. We explore implications for combinatorial optimization problems pertaining to sparse systems.

A novel Soft Actor–Critic framework with disjunctive graph embedding and autoencoder mechanism for Job Shop Scheduling Problems

The Job-Shop Scheduling Problem (JSSP) is a well-established and classic NP-hard combinatorial optimization issue. The quality of its scheduling scheme directly affects the operational efficiency of manufacturing systems. Priority Dispatching Rules (PDRs) are often utilized to address JSSP in real-world contexts, but the process of creating effective PDRs can be daunting and time-consuming. It also necessitates comprehensive domain knowledge, typically resulting in mediocre performance. In this paper, we introduce a novel reinforcement learning (RL) model called Disjunctive Graph Embedding with Autoencoder Mechanism for Job Shop Scheduling Problems (DGEAM-JSSP), designed to automate PDRs learning. Our proposed model confronts the issue using a Graph Neural Network (GNN) to learn node features that encapsulate the spatial structure of the JSSP graph representation. The ensuing policy network is size-agnostic, enabling effective generalization on larger-scale instances. Additionally, we employ a transformer encoder, incorporating parallel encoding and a self-attention mechanism, to successfully recognize long-term dependencies among operations in large-scale scheduling problems. We also implemented an end-to-end training approach using the Soft Actor–Critic (SAC) algorithm to instruct the two modules. Computational experiment results reveal that, with a single training, our agent successfully learns a superior dispatching policy, surpassing PDRs and state-of-the-art RL frameworks specifically tailored for each JSSP instance size in solution quality, as well as OR-Tools in execution speed. Moreover, results from random and benchmark instances illustrate that the uniquely-modeled learned policies have impressive generalization performance on real-world instances and significantly larger-scale scenarios involving up to 2000 operations.

Variational Quantum Algorithm‐Preserving Feasible Space for Solving the Uncapacitated Facility Location Problem

AbstractThe Quantum Alternating Operator Ansatz (QAOA+) is one of the Variational Quantum Algorithm (VQA) specifically developed to tackle combinatorial optimization problems by exploring the feasible space in search of a target solution. For the Constrained Binary Optimization with Unconstrained Variables Problems (CBO‐UVPs), the mixed operators in the QAOA+ circuit are applied to the constrained variables, while the single‐qubit rotating gates operate on the unconstrained variables. The expressibility of this circuit is limited by the shortage of two‐qubit gates and the parameter sharing in the single‐qubit rotating gates, which consequently impacts the performance of QAOA+ for solving CBO‐UVPs. Therefore, it is crucial to develop a suitable ansatz for CBO‐UVPs. In this paper, the Variational Quantum Algorithm‐Preserving Feasible Space (VQA‐PFS) ansatz is proposed, exemplified by the Uncapacitated Facility Location Problem (UFLP), that applies mixed operators on constrained variables while employing Hardware‐Efficient Ansatz (HEA) on unconstrained variables. The numerical results demonstrate that VQA‐PFS significantly enhances the probability of success and exhibits faster convergence than QAOA+, Quantum Approximation Optimization Algorithm (QAOA), and HEA. Furthermore, VQA‐PFS reduces the circuit depth dramatically compared to QAOA+ and QAOA. The algorithm is general and instructive in tackling CBO‐UVPs.

A Comparison of Heuristic Algorithms for Solving the Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a challenging computational problem in combinatorial optimization that aims to visit all cities exactly once and return to the first city. Despite that numerous theoretical solutions have been proposed in the literature, finding the exact optimal solution remains computationally infeasible due to the NP-hard nature of the problem. To address this, many heuristic and optimization approaches have been developed to generate probabilistic results that are often approximations. This paper presents a comparison between four popular algorithms: steepest ascent hill climbing, simulated annealing, genetic algorithm with partially matched crossover, and Particle Swarm Optimization (PSO). The study examines how these algorithms can solve the TSP and avoid local minimum values while achieving a balance between research exploration and exploitation for an optimal solution. For a relatively large number of cities, the simulated annealing algorithm and genetic algorithm produce promising results whilst the genetic algorithm takes longer time to execute due to the iterative application of its variation operators.

A Graph-Neural-Network-Powered Solver Framework for Graph Optimization Problems.

Backtracking combined with branching heuristics is a prevalent approach for tackling constraint satisfaction problems (CSPs) and combinatorial optimization problems (COPs). While branching heuristics specifically designed for certain problems can be theoretically efficient, they are often complex and difficult to implement in practice. On the other hand, general branching heuristics can be applied across various problems, but at the risk of suboptimality. We introduce a solver framework that leverages the Shannon entropy in branching heuristics to bridge the gap between generality and specificity in branching heuristics. This enables backtracking to follow the path of least uncertainty, based on probability distributions that conform to problem constraints. We employ graph neural network (GNN) models with loss functions derived from the probabilistic method to learn these probability distributions. We have evaluated our approach by its applications to two NP-hard problems: the (minimum) dominating-clique problem and the edge-clique-cover problem. Compared with the state-of-the-art solvers for both problems, our solver framework outputs competitive results. Specifically, for the (minimum) dominating-clique problem, our approach generates fewer branches than the solver presented by Culberson et al. (2005). For the edge-clique-cover problem, our approach produces smaller-sized edge clique covers (ECCs) than the solvers referenced by Conte et al. (2020) and Kellerman (1973).

Design and Implementation of a VLSI-based Annealing Accelerator for Efficiently Solving Combinatorial Optimization Problems

Design and Implementation of a VLSI-based Annealing Accelerator for Efficiently Solving Combinatorial Optimization Problems

Optimal Launch Manifesting for Space Freight Forwarders

This paper proposes a novel framework to optimize a launch vehicle manifesting plan for small satellites from the perspective of a space freight forwarder. The forwarder in space logistics coordinates individual shipment orders from the launch customers and launch service providers like a freight forwarder in ground logistics does. An optimal launch manifesting problem is formulated as a variant of the vehicle routing problem involving the decisions on carriers, launch strategies (dedicated or multi-payload), launch dates, and satellite deployment sequences. A solution procedure based on the column generation method is proposed to address the challenges associated with the large-scale combinatorial optimization problem. A case study demonstrates the effectiveness of the proposed framework and solution procedure.

Optimizing link-length fitting between an operator and a robot with compensation for habitual movements

Teleoperation has become increasingly important for the real-world application of robots. However, in current teleoperation applications, operators need to visually recognize any physical deviations between the robot and themselves and correct its operation accordingly. Even when humans move their bodies, achieving high positional accuracy can be difficult for them, which can limit their speed of movement and place a heavy burden on them. In this study, we proposed a parameter optimization method for feedforward compensation of the link-length deviation and operator's habitual body movements in leader–follower control in 3D space. To optimize the parameters, we used Digital Annealer developed by Fujitsu Ltd, which could rapidly solve the combinatorial optimization problem. The objective function minimized the difference between the hand positions of the robot and targets. Simulations verified that the proposed method could reduce hand positional differences.

Multiobjective optimization of collaborative robotic task sequence assignment problems under collision-free constraints

This paper proposes a multiobjective optimization approach to address the challenge of collaborative manufacturing with multiple robot arms. Given the necessity for multiple robot arms to work together, the potential for collisions between robotic motions is a significant concern, and the automated task sequence assignment for robots becomes increasingly complex. Previous research has either simplified the collision-free conditions in a limited working area, or employed a master-slave approach to obtain only a local solution. Consequently, we propose a unified global optimization approach for simultaneously addressing various collaborative manufacturing issues, including robotic task sequence assignment (RTSA), multiple inverse kinematics (IK) selection, joint-space collision-free operations and multiple manufacturing objectives. As the optimal collaborative RTSA problem is a combinatorial optimization problem with non-deterministic polynomial-time hard (NP-hard) complexity, this paper presents a hybrid nondominated sorting genetic algorithm III (NSGA-III) method that integrates a Hamming-distance-based method and a greedy strategy within NSGA-III to improve population diversity and solution quality. To validate the efficacy of the proposed approach, simulation experiments were conducted on cooperative manufacturing scenarios, with two objectives: task completion time and task load balancing. The experimental results demonstrate that the proposed approach is effective in obtaining collision-free Pareto solutions. Furthermore, the proposed hybrid NSGA-III method obtains superior solutions compared to the original method in the studied problem, as measured by two performance indices.

Machine Learning based Algorithm Selection and Genetic Algorithms for serial-batch scheduling

Whenever combinatorial optimization problems cannot be solved by exact solution methods in reasonable time, tailor-made algorithms (heuristics, meta-heuristics) are developed. Often, these heuristics exploit structural properties and perform well on selected subsets of the problem space. For example, this is how the two best-known construction heuristics solve the scheduling problem investigated in this study (i.e., the scheduling of parallel serial-batch processing machines with incompatible job families, restricted batch capacities, arbitrary batch capacity demands, and sequence-dependent setup times). However, when the properties change, the performance of one algorithm might decrease, and another algorithm might have been the better choice. To resolve this issue, we propose using Machine Learning to exploit the strengths of different algorithms and to select the probably best-performing algorithm for each problem instance individually. To that, we investigate a variety of methods from the “learning-to-rank” literature and propose several adaptations. Furthermore, because there is no algorithm for the considered scheduling problem that is capable to explore the entire solution space, we developed two Genetic Algorithms for the improvement of initial solutions computed by the selected algorithms. Here, we put special emphasis on ensuring that the solution representation (encoding) reflects the entire solution space and that the operators (e.g., for recombination and mutation) are appropriate to explore and exploit this space completely. Our computational experiments show an average increase of 39.19% in solution quality.

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.

Optimized Selective Assembly using Hungarian Algorithm

Assembly of discrete parts guided by hardware design specification constitutes the final phase in product manufacturing. In the course of mass production of components, mating parts with geometric or dimensional deviation from their intended design can be made acceptable by the identification of suitable pairs after analysing design fit and tolerance limits. By transforming this application-specific problem into a unified mathematical model, an optimal solution can be achieved that minimizes the rejection of non-conforming fabricated parts. Regardless of the type and range of a design fit, the problem can be mapped into a matrix using a ranking function defined by the user. The ranking function is modifiable as per the user requirements and may vary based on the selection criteria for an assembly. Based on the type of ranking function used, the tabulated matrix is solved using the Hungarian minimization/maximization algorithm, which is a powerful combinatorial optimization algorithm that solves the classical assignment problem in mathematics. This approach ensures maximum number of suiting pairs as well as nominal suiting of parts with each other resulting in high-quality products and maximum utilization of fabricated resources.