Combinatorial Optimization Research Articles

A matheuristic-oriented iterated greedy algorithm for multi-mode resource-constrained project scheduling problem under uncertainty

Multi-mode resource-constrained project scheduling problem is one of the most important branches of combinatorial optimization problems. The aim is to find a feasible solution consisting of activity start time and execution mode so that the project makespan is minimized. However, there are a lot of uncertain events, that bring great challenges to project scheduling. We first propose a mixed integer linear programming model where the activity duration is uncertain. Then two uncertain parameters are introduced to describe the disturbance degree of uncertain activity duration and the allowable violation degree of constraints respectively, and the MILP model with uncertain activity duration is converted to its deterministic robust counterpart model. A matheuristic local optimization approach is proposed to balance computational time with the optimization of the computational result. A matheuristic-oriented iterated greedy algorithm that combines a matheuristic local optimization approach and an iterated greedy algorithm is proposed to solve this problem in this paper. Experimental results indicate that the robust counterpart model can obtain robust optimal solutions for small-scale instances in accepted computational time. Meanwhile, the proposed matheuristic-oriented iterated greedy algorithm can obtain robust near-optimal solutions for large-scale instances and is superior to other compared algorithms.

A benchmarking study of quantum algorithms for combinatorial optimization

We study the performance scaling of three quantum algorithms for combinatorial optimization: measurement-feedback coherent Ising machines (MFB-CIM), discrete adiabatic quantum computation (DAQC), and the Dürr–Høyer algorithm for quantum minimum finding (DH-QMF) that is based on Grover’s search. We use MaxCut problems as a reference for comparison, and time-to-solution (TTS) as a practical measure of performance for these optimization algorithms. For each algorithm, we analyze its performance in solving two types of MaxCut problems: weighted graph instances with randomly generated edge weights attaining 21 equidistant values from −1 to 1; and randomly generated Sherrington–Kirkpatrick (SK) spin glass instances. We empirically find a significant performance advantage for the studied MFB-CIM in comparison to the other two algorithms. We empirically observe a sub-exponential scaling for the median TTS for the MFB-CIM, in comparison to the almost exponential scaling for DAQC and the proven Õ2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{{{{\\mathcal{O}}}}}\\left(\\sqrt{{2}^{n}}\\right)$$\\end{document} scaling for DH-QMF. We conclude that the MFB-CIM outperforms DAQC and DH-QMF in solving MaxCut problems.

Train dispatching program for high-speed railway station based on genetic algorithm

In case of train delays, centralized traffic control system become disabled, and the workload of dispatchers increases dramatically. Based on genetic algorithm, the author designs a program to appropriately reschedule trains in terms of delays, minimizing the total delay time and changes of gate. The author transformed the initial problem to a compromised combinatorial optimization model, with total delay time, changes of gate and conflicting routes as objectives. The high weighting in conflicting routes ensures efficiency and high probability of obtaining a feasible solution. With discreate variants, the author designs special coding and evolving method suitable for this problem. Using a special treatment for conflicts and initializing chromosomes, the program can construct new timetable quickly given the scheduled timetable, predicted arrival time and order of trains (optional), which promotes the efficiency and security of dispatching in high-speed railway stations. The method was tested with a synthetic data of Shanghai-Kunming section of Hangzhou East Railway Station.

Rydberg‐Atom Graphs for Quadratic Unconstrained Binary Optimization Problems

AbstractThere is a growing interest in harnessing the potential of the Rydberg‐atom system to address complex combinatorial optimization challenges. Here an experimental demonstration of how the quadratic unconstrained binary optimization (QUBO) problem can be effectively addressed using Rydberg‐atom graphs is presented. The Rydberg‐atom graphs are configurations of neutral atoms organized into mathematical graphs, facilitated by programmable optical tweezers, and designed to exhibit many‐body ground states that correspond to the maximum independent set (MIS) of their respective graphs. Four elementary Rydberg‐atom subgraph components are developed, not only to eliminate the need of local control but also to be robust against interatomic distance errors, while serving as the building blocks sufficient for formulating generic QUBO graphs. To validate the feasibility of the approach, a series of Rydberg‐atom experiments selected to demonstrate proof‐of‐concept operations of these building blocks are conducted. These experiments illustrate how these components can be used to programmatically encode the QUBO problems to Rydberg‐atom graphs and, by measuring their many‐body ground states, how their QUBO solutions are determined subsequently.

Variational Quantum Algorithms for Semidefinite Programming

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., N≫M, where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

Using Constraint Propagation to Bound Linear Programs

We present an approach to compute bounds on the optimal value of linear programs based on constraint propagation. Given a feasible dual solution, we apply constraint propagation to the complementary slackness conditions and, if propagation succeeds to prove these conditions infeasible, the infeasibility certificate (in the sense of Farkas’ lemma) is reconstructed from the propagation history. This certificate is a dual-improving direction, which allows us to improve the bound. As constraint propagation need not always detect infeasibility of a linear inequality system, the method is not guaranteed to converge to a global solution of the linear program but only to an upper bound, whose tightness depends on the structure of the program and the used propagation method. The approach is suited for large sparse linear programs (such as LP relaxations of combinatorial optimization problems), for which the classical LP algorithms may be infeasible, if only for their super-linear space complexity. The approach can be seen as a generalization of the Virtual Arc Consistency (VAC) algorithm to bound the LP relaxation of the Weighted CSP (WCSP). We newly apply it to the LP relaxation of the Weighted Max-SAT problem, experimentally showing that the obtained bounds are often not far from optima of the relaxation and proving that they are exact for known tractable subclasses of Weighted Max-SAT.

Bi-Objective Combinatorial Optimization Model for Emission Reduction Projects at Container Terminals Considering Investment Amount and Reduction Efficiency

Bi-Objective Combinatorial Optimization Model for Emission Reduction Projects at Container Terminals Considering Investment Amount and Reduction Efficiency

Towards Adiabatic Quantum Computing Using Compressed Quantum Circuits

We describe tensor network algorithms to optimize quantum circuits for adiabatic quantum computing. To suppress diabatic transitions, we include counterdiabatic driving in the optimization and utilize variational matrix product operators to represent adiabatic gauge potentials. Traditionally, Trotter product formulas are used to turn adiabatic time evolution into quantum circuits and the addition of counterdiabatic driving increases the circuit depth per time step. Instead, we classically optimize a parameterized quantum circuit of fixed depth to simultaneously capture adiabatic evolution together with counterdiabatic driving over many time steps. The methods are applied to the ground-state preparation of quantum Ising chains with transverse and longitudinal fields. We show that the classically optimized circuits can significantly outperform Trotter product formulas. Additionally, we discuss how the approach can be used for combinatorial optimization. Published by the American Physical Society 2024

Accelerated Driving-Training-Based Optimization for Solving Constrained Bi-Objective Stochastic Optimization Problems

The constrained bi-objective stochastic optimization problem (CBSOP) considers the optimization problem with stochastic bi-objective functions subject to deterministic constraints. The CBSOP is part of a set of hard combinatorial optimization problems regarding time complexity. Ordinal optimization (OO) theory provides a commonly recognized structure to handle hard combinatorial optimization problems. Although OO theory may solve hard combinatorial optimization problems quickly, the deterministic constraints will critically influence computing performance. This work presents a metaheuristic approach that combines driving-training-based optimization (DTBO) with ordinal optimization (OO), abbreviated as DTOO, to solve the CBSOP with a large design space. The DTOO approach comprises three major components: the surrogate model, diversification, and intensification. In the surrogate model, the regularized minimal-energy tensor product with cubic Hermite splines is utilized as a fitness estimation of design. In diversification, an accelerated driving-training-based optimization is presented to determine N remarkable designs from the design space. In intensification, a reinforced optimal computing budget allocation is used to find an extraordinary design from the N remarkable designs. The DTOO approach is applied to a medical resource allocation problem in the emergency department. Simulation results obtained by the DTOO approach are compared with three heuristic approaches to examine the performance of the DTOO approach. Test results show that the DTOO approach obtains an extraordinary design with higher solution quality and computational efficiency than the three heuristic approaches.

Location and routing of armed Unmanned Aerial Vehicles and carrier platforms against mobile targets

In this study, we consider a real-life combinatorial optimization problem related to deploying and routing Unmanned Aerial Vehicles (UAVs) and naval carrier platforms. In particular, we seek to determine the initial locations for carrier platforms and the optimal type and number of UAVs to be stationed on each carrier platform as well as their spatial/temporal routes for engaging hostile surface targets in the region. Our modeling framework incorporates a number of realistic but challenging ingredients and assumptions such as the mobility of surface targets and carrier platforms during the mission, capacitated multiple platforms and UAVs, UAV-carrier platform compatibility, and allowance for different takeoff/land on platforms for UAVs. In the effort to represent the problem mathematically, we first formulated an Integer Linear Program (ILP) model which seeks to maximize the total time-dependent weights of the targets engaged. Next, we proposed a heuristic solution algorithm based on the ant colony optimization framework. Our computational experiments performed on instances with different sizes showed that the heuristic approach achieves high-quality solutions even for large-size problem instances in short CPU times.

Shape optimization of autonomous underwater helicopters based on different parameter curves and various optimization algorithms

This study aimed to compare the non-uniform rational B-spline (NURBS) and Bezier curves to design the shape of an autonomous underwater helicopter (AUH). Currently, there is no established method for AUH shape design. The optimal solutions obtained from both curves were compared to evaluate their performance. A parametric model was constructed to determine the design variables of the two curves. Computational fluid dynamics was used to solve the resistance values of different shapes. Sample points were generated using the design of the experimental approach, and an approximate model was established using the radial basis function neural network method. Based on the combinatorial optimization strategy, three intelligent optimization algorithms were used for the global search. Then, the resulting solution was used as the starting point for a local search to obtain the global optimal solution. The resistance optimal solutions achieved by employing the NURBS and Bezier curves were 25.2% and 24.1% smaller than the initial values, respectively. These findings indicate that, within the design parameters of this study, the shape design approach using the NURBS curve yields superior performance for the AUH.

A comprehensive survey on the chicken swarm optimization algorithm and its applications: state-of-the-art and research challenges

The application of optimization theory and the algorithms that are generated from it has increased along with science and technology's continued advancement. Numerous issues in daily life can be categorized as combinatorial optimization issues. Swarm intelligence optimization algorithms have been successful in machine learning, process control, and engineering prediction throughout the years and have been shown to be efficient in handling combinatorial optimization issues. An intelligent optimization system called the chicken swarm optimization algorithm (CSO) mimics the organic behavior of flocks of chickens. In the benchmark problem's optimization process as the objective function, it outperforms several popular intelligent optimization methods like PSO. The concept and advancement of the flock optimization algorithm, the comparison with other meta-heuristic algorithms, and the development trend are reviewed in order to further enhance the search performance of the algorithm and quicken the research and application process of the algorithm. The fundamental algorithm model is first described, and the enhanced chicken swarm optimization algorithm based on algorithm parameters, chaos and quantum optimization, learning strategy, and population diversity is then categorized and summarized using both domestic and international literature. The use of group optimization algorithms in the areas of feature extraction, image processing, robotic engineering, wireless sensor networks, and power. Second, it is evaluated in terms of benefits, drawbacks, and application in comparison to other meta-heuristic algorithms. Finally, the direction of flock optimization algorithm research and development is anticipated.

Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem

The split delivery vehicle routing problem (SDVRP) is a classic combinatorial optimization problem, which is usually solved using a heuristic algorithm. The ant colony optimization algorithm is an excellent heuristic algorithm that has been successfully applied to solve various practical problems, and it has achieved good results. However, in the existing ant colony optimization algorithms, there are issues with weak targeting of different customer selection strategies, difficulty in balancing convergence speed and global search ability, and a predisposition to become trapped in local optima. To solve these problems, this paper proposes an improved ant colony algorithm (IACA). First, in terms of customer point selection, the initial customer and noninitial customer selection strategies are proposed for different customers, and the adaptive selection threshold is designed. Second, in terms of pheromone processing, an initial pheromone distribution method based on a greedy strategy, a pheromone backtracking mechanism, and an adaptive pheromone volatile factor are proposed. Finally, based on the 2-opt local search method, vehicle path self-search and intervehicle path search are proposed to further improve the quality of the solution. This paper tests the performance of the IACA on datasets of different scales. The experimental results show that compared with the clustering algorithm, artificial bee colony algorithm, particle swarm optimization algorithm, traditional ant colony algorithm, and other algorithms, the IACA can achieve more competitive results. Specifically, compared to the path length calculated by other algorithms, the path length calculated by IACA decreased by an average of 1.58%, 4.28%, and 3.64% in small, medium, and large-scale tests, respectively.

Faster Counting and Sampling Algorithms using Colorful Decision Oracle

In this work, we consider d - Hyperedge Estimation and d - Hyperedge Sample problems that deal with estimation and uniform sampling of hyperedges in a hypergraph ℋ( U (ℋ), ℱ(ℋ) in the query complexity framework, where U (ℋ) denotes the set of vertices and ℱ(ℋ) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle ( CID ), which takes d (non-empty) pairwise disjoint subsets of vertices A 1 ,..., A d ⊆ U (ℋ) as input and answers whether there exists a hyperedge in ℋ having exactly one vertex in each A i for all i ∈ {1, 2, ..., d }. Apart from the fact that d - Hyperedge Estimation and d - Hyperedge Sample problems with CID oracle access seem to be nice combinatorial problems, Dell et al. [SODA’20 & SICOMP’22] established that decision vs. counting complexities of a number of combinatorial optimization problems can be abstracted out as d - Hyperedge Estimation problem with a CID oracle access. The main technical contribution of this article is an algorithm that estimates m = |ℱ(ℋ)| with \(\widehat{m}\) such that \(\begin{equation*} \frac{1}{C_{d}\log ^{d-1} n} \;\le \; \frac{\widehat{m}}{m} \;\le \; C_{d} \log ^{d-1} n \end{equation*}\) by using at most C d log d +2 n CID queries, where n denotes the number of vertices in the hypergraph ℋ and C d is a constant that depends only on d . Our result, when coupled with the framework proposed by Dell et al. (SODA’20 & SICOMP’22), leads to implies improved bounds for (1 ± ε)-approximation (where ε ∈ (0,1)) for the following fundamental problems: Edge Estimation using the Bipartite Independent Set ( BIS ) query. We improve the bound obtained by Beame et al. (ITCS’18 & TALG’20). Triangle Estimation using the Tripartite Independent Set ( TIS ) query. Currently, Dell et al.’s result gives the best bound for the case of triangle estimation in general graphs (SODA’20 & SICOMP’22). The previous best bound for the case of graphs with low co-degree (co-degree of a graph is the maximum number of triangles incident over any edge of the graph) was due to Bhattacharya et al. (ISAAC’19 & TOCS’21). We improve both of these bounds. Hyperedge Estimation & Sampling using Colorful Independence Oracle ( CID ). We give an improvement over the bounds obtained by Dell et al. (SODA’20 & SICOMP’22).

ILP-based resource optimization realized by quantum annealing for optical wide-area communication networks—A framework for solving combinatorial problems of a real-world application by quantum annealing

Resource allocation of wide-area internet networks is inherently a combinatorial optimization problem that if solved quickly, could provide near real-time adaptive control of internet-protocol traffic ensuring increased network efficacy and robustness, while minimizing energy requirements coming from power-hungry transceivers. In recent works we demonstrated how such a problem could be cast as a quadratic unconstrained binary optimization (QUBO) problem that can be embedded onto the D-Wave Advantage™ quantum annealer system, demonstrating proof of principle. Our initial studies left open the possibility for improvement of D-Wave solutions via judicious choices of system run parameters. Here we report on our investigations for optimizing these system parameters, and how we incorporate machine learning (ML) techniques to further improve on the quality of solutions. In particular, we use the Hamming distance to investigate correlations between various system-run parameters and solution vectors. We then apply a decision tree neural network (NN) to learn these correlations, with the goal of using the neural network to provide further guesses to solution vectors. We successfully implement this NN in a simple integer linear programming (ILP) example, demonstrating how the NN can fully map out the solution space that was not captured by D-Wave. We find, however, for the 3-node network problem the NN is not able to enhance the quality of space of solutions.

Analysis and Study of Transmission Line Icing Based on Grey Correlation Pearson Combinatorial Optimization Support Vector Machine

In recent years, ice and snow disasters have occurred frequently, and transmission line icing can pose a great threat to the operation of the power system. In order to solve this problem, the power industry should introduce an intelligent monitoring system to recognize the risk of ice-covered disasters in advance and take timely countermeasures to reduce the impact of extreme weather on the safe operation of the power system. In this paper, we first use Python to sort out and summarize multidimensional meteorological data from multiple locations in the Northeast China, and propose a method called Grey Relational Pearson Combination Analysis (GPCA). Through this method, we determine the correlation between multidimensional meteorological data and icing degree, and select the optimal combination of icing factors as the feature vector for a support vector machine. In addition, we also compare the differences in classification effects of four types of kernel functions, such as linear kernel, polynomial kernel and Gaussian kernel. Finally, considering data balance, we use Gaussian kernel to establish GPCA-SVM model and conduct experimental analysis. There was a 1.04% increase in accuracy, a 1.36% increase in recall, and 0.31% increase in F1 value when comparing gray correlation accuracy and a 21.66% increase in precision when comparing Pearson accuracy at light icing. Comparison of gray correlation accuracy at heavy icing improved by 1.04%, precision by 17.44%, and F1 value by 10.29%; comparison of Pearson accuracy improved by 0.31%, precision by 9.93%, and F1 value by 1.67%. These research results lay the foundation for the intelligent monitoring system of power grid, and provide guarantee for the operation and maintenance of power system.

A multi-strategy driven reinforced hierarchical operator in the grey wolf optimizer for feature selection

In this study, a multi-strategy driven reinforced hierarchical operator for a grey wolf optimizer (RHGWO) is proposed to solve the feature selection (FS) problem, whereby tedious data are converted into information and often modeled as a combinatorial optimization problem. First, a multi-strategy mechanism is proposed to provide the GWO algorithm with exploration capabilities, including memory-based diversity and Lévy flight-based extension search. Next, a hierarchical segmentation technique is proposed to allocate exploration and exploitation, thereby providing exploration capability for superior wolves to search diverse regions and exploitation capability for inferior wolves to converge to the promising area. Subsequently, a chaotic elite learning strategy is designed for leaders to prevent misdirection. Finally, a more rational nonlinear parameter transformation is designed. Multiple experiments validate the adaptability and versatility of the proposed RHGWO algorithm.

Minimum connected dominating set and backbone of a random graph

We study the minimum dominating set problem as a representative combinatorial optimization challenge with a global topological constraint. The requirement that the backbone induced by the vertices of a dominating set should be a connected subgraph makes the problem rather nontrivial to investigate by statistical physics methods. Here, we convert this global connectivity constraint into a set of local vertex constraints and build a spin glass model with only five coarse-grained vertex states. We derive a set of coarse-grained belief-propagation equations and obtain theoretical predictions of the relative sizes of the minimum dominating sets for regular random and Erdös–Rényi random graph ensembles. We also implement an efficient message-passing algorithm to construct close-to-minimum connected dominating sets and backbone subgraphs for single random graph instances. Our theoretical strategy may also be applicable to some other global topological constraints.

Scalable Primal Heuristics Using Graph Neural Networks for Combinatorial Optimization

By examining the patterns of solutions obtained for various instances, one can gain insights into the structure and behavior of combinatorial optimization (CO) problems and develop efficient algorithms for solving them. Machine learning techniques, especially Graph Neural Networks (GNNs), have shown promise in parametrizing and automating this laborious design process. The inductive bias of GNNs allows for learning solutions to mixed-integer programming (MIP) formulations of constrained CO problems with a relational representation of decision variables and constraints. The trained GNNs can be leveraged with primal heuristics to construct high-quality feasible solutions to CO problems quickly. However, current GNN-based end-to-end learning approaches have limitations for scalable training and generalization on larger-scale instances; therefore, they have been mostly evaluated over small-scale instances. Addressing this issue, our study builds on supervised learning of optimal solutions to the downscaled instances of given large-scale CO problems. We introduce several improvements on a recent GNN model for CO to generalize on instances of a larger scale than those used in training. We also propose a two-stage primal heuristic strategy based on uncertainty-quantification to automatically configure how solution search relies on the predicted decision values. Our models can generalize on 16x upscaled instances of commonly benchmarked five CO problems. Unlike the regressive performance of existing GNN-based CO approaches as the scale of problems increases, the CO pipelines using our models offer an incremental performance improvement relative to CPLEX. The proposed uncertainty-based primal heuristics provide 6-75% better optimality gap values and 45-99% better primal gap values for the 16x upscaled instances and brings immense speedup to obtain high-quality solutions. All these gains are achieved through a computationally efficient modeling approach without sacrificing solution quality.

A heuristic method for solving the Steiner tree problem in graphs using network centralities

We propose a heuristic method of using network centralities for constructing small-weight Steiner trees in this paper. The Steiner tree problem in graphs is one of the practical NP-hard combinatorial optimization problems. Given a graph and a set of vertices called terminals in the graph, the objective of the Steiner tree problem in graphs is to find a minimum weight Steiner tree that is a tree containing all the terminals. Conventional construction methods make a Steiner tree based on the shortest paths between terminals. If these shortest paths are overlapped as much as possible, we can obtain a small-weight Steiner tree. Therefore, we proposed to use network centralities to distinguish which edges should be included to make a small-weight Steiner tree. Experimental results revealed that using the vertex or the edge betweenness centralities contributes to making small-weight Steiner trees.