Hard Combinatorial Optimization Problems Research Articles

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.

Application of Lexisearch Algorithm to Vehicle Routing Problem with Time Windows

Vehicle Routing Problem with Time Windows (VRPTW) is an NP hard combinatorial scheduling optimization problem in which a minimum number of routes have to be determined so as to serve each of the destinations within their specified time windows. In this paper, the mail delivery of the Nigerian Postal Services (NIPOST) is modelled as a VRPTW in order to address the problem of delay in mail delivery occurring regularly in NIPOST. The Abuja Post Office is used as a case study and the Model of a related literature is applied with modifications to solve the problem. The problem is solved by applying Lexisearch algorithm using data that was obtained from Abuja Post Office and computational results on Solomon’s 100 instances were used to validate the algorithm.

Comparing Adiabatic Quantum Computers for satellite images feature extraction

Adiabatic Quantum Computers (AQCs) are special-purpose devices that promise to speed up the resolution of hard combinatorial optimization problems by exploiting quantum mechanical phenomena. Despite representing one of the most mature quantum computational paradigms, AQCs are often benchmarked on spin-glass problems that conveniently mimic their internal structure, and their performances are usually compared to those of suboptimal Simulated Annealing solvers. In this work, we evaluate the capabilities of AQCs to extract features from a dataset of low-resolution satellite images of airplanes, exploiting an approach based on matrix factorization. We assess the performance of three generations of quantum devices provided by the D-Wave company by analyzing their behavior via selected evaluation metrics as a function of the problem size. We also compare against classical results obtained with a commercial solver. Additionally, we outline a parameter-tuning procedure that allows for harnessing the full potential of AQCs. Although the quantum devices still tend to underperform their classical counterparts, we observe the two most recent AQCs exhibiting superior performance for the tested instances with the smallest size. Additionally, our results indicate incremental performance improvements across new AQCs generations. These observations suggest cautious optimism about the potential future applicability of such computational tools.

Efficient Dynamic Distributed Resource Slicing in 6G Multi-Access Edge Computing Networks With Online ADMM and Message Passing Graph Neural Networks

We consider the problem of resource slicing in the 6 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> generation multi-access edge computing (6G-MEC) network. The network includes many non-stationary space-air-ground-sea nodes with dynamic, unstable connections and resources, where any node can be in one of two hidden states: i) reliable – when the node generates/propagates no data errors; ii) unreliable – when the node can generate/propagate random errors. We show that solving this problem is challenging, since it represents a non-deterministic polynomial-time (NP) hard dynamic combinatorial optimization problem depending on the unknown distribution of hidden nodes' states and time-varying parameters (connections and resources of nodes) which can only be observed locally. To tackle these challenges, we develop a new deep learning (DL) model based on the message passing graph neural network (MPNN) to estimate hidden nodes' states in dynamic network environments. We then propose a novel algorithm based on the integration of MPNN-based DL and online alternating direction method of multipliers (ADMM) – extension of the well-known classical “static” ADMM to dynamic settings, where the slicing problem is solved distributedly, in real time, based on local information. We prove that our algorithm converges to a global optimum of our problem with a superior performance even in the highly-dynamic, unreliable scenarios.

Ferroelectric compute-in-memory annealer for combinatorial optimization problems

Computationally hard combinatorial optimization problems (COPs) are ubiquitous in many applications. Various digital annealers, dynamical Ising machines, and quantum/photonic systems have been developed for solving COPs, but they still suffer from the memory access issue, scalability, restricted applicability to certain types of COPs, and VLSI-incompatibility, respectively. Here we report a ferroelectric field effect transistor (FeFET) based compute-in-memory (CiM) annealer for solving larger-scale COPs efficiently. Our CiM annealer converts COPs into quadratic unconstrained binary optimization (QUBO) formulations, and uniquely accelerates in-situ the core vector-matrix-vector (VMV) multiplication operations of QUBO formulations in a single step. Specifically, the three-terminal FeFET structure allows for lossless compression of the stored QUBO matrix, achieving a remarkably 75% chip size saving when solving Max-Cut problems. A multi-epoch simulated annealing (MESA) algorithm is proposed for efficient annealing, achieving up to 27% better solution and ~ 2X speedup than conventional simulated annealing. Experimental validation is performed using the first integrated FeFET chip on 28nm HKMG CMOS technology, indicating great promise of FeFET CiM array in solving general COPs.

Neural network to solve fuzzy constraint satisfaction problems

&lt;p&gt;It has been proven that solving the constraint satisfaction problem (CSP) is an No Polynomial hard combinatorial optimization problem. This holds true even in cases where the constraints are fuzzy, known as fuzzy constraint satisfaction problems (FCSP). Therefore, the continuous Hopfield neural network model can be utilized to resolve it. The original algorithm was developed by Talaavan in 2005. Many practical problems can be represented as a FCSP. In this paper, we expand on a neural network technique that was initially developed for solving CSP and adapt it to tackle problems that involve at least one fuzzy constraint. To validate the enhanced effectiveness and rapid convergence of our proposed approach, a series of numerical experiments are carried out. The results of these experiments demonstrate the superior performance of the new method. Additionally, the experiments confirm its fast convergence. Specifically, our study focuses on binary instances with ordinary constraints to test the proposed resolution model. The results confirm that both the proposed approaches and the original continuous Hopfield neural network approach exhibit similar performance and robustness in solving ordinary constraint satisfaction problems.&lt;/p&gt;

Adaptive population-based simulated annealing for resource constrained job scheduling with uncertainty

Transporting ore from mines to ports is of significant interest in mining supply chains. These operations are commonly associated with growing costs and a lack of resources. Large mining companies are interested in optimally allocating resources to reduce operational costs. This problem has been previously investigated as resource constrained job scheduling (RCJS). While a number of optimisation methods have been proposed to tackle the deterministic problem, the uncertainty associated with resource availability, an inevitable challenge in mining operations, has received less attention. RCJS with uncertainty is a hard combinatorial optimisation problem that is challenging for existing optimisation methods. We propose an adaptive population-based simulated annealing algorithm that can overcome existing limitations of methods for RCJS with uncertainty, including pre-mature convergence, excessive number of hyper-parameters, and the inefficiency in coping with different uncertainty levels. This new algorithm effectively balances exploration and exploitation, by using a population, modifying the cooling schedule in the Metropolis-Hastings algorithm, and using an adaptive mechanism to select perturbation operators. The results show that the proposed algorithm outperforms existing methods on a benchmark RCJS dataset considering different uncertainty levels. Moreover, new best known solutions are discovered for all but one problem instance across all uncertainty levels.

Reinforcement learning assisted recursive QAOA

In recent years, variational quantum algorithms such as the Quantum Approximation Optimization Algorithm (QAOA) have gained popularity as they provide the hope of using NISQ devices to tackle hard combinatorial optimization problems. It is, however, known that at low depth, certain locality constraints of QAOA limit its performance. To go beyond these limitations, a non-local variant of QAOA, namely recursive QAOA (RQAOA), was proposed to improve the quality of approximate solutions. The RQAOA has been studied comparatively less than QAOA, and it is less understood, for instance, for what family of instances it may fail to provide high-quality solutions. However, as we are tackling NP-hard problems (specifically, the Ising spin model), it is expected that RQAOA does fail, raising the question of designing even better quantum algorithms for combinatorial optimization. In this spirit, we identify and analyze cases where (depth-1) RQAOA fails and, based on this, propose a reinforcement learning enhanced RQAOA variant (RL-RQAOA) that improves upon RQAOA. We show that the performance of RL-RQAOA improves over RQAOA: RL-RQAOA is strictly better on these identified instances where RQAOA underperforms and is similarly performing on instances where RQAOA is near-optimal. Our work exemplifies the potentially beneficial synergy between reinforcement learning and quantum (inspired) optimization in the design of new, even better heuristics for complex problems.

Genetic Algorithm for Solving Mobile Robot Scheduling Problem in Flexible Manufacturing Environment

The scheduling genuinely a complex process, aimed at optimizing operational activities in pursuit of one or more objectives by leveraging production data which may include previous schedules. The scheduling problem in Flexible Manufacturing System (FMS) is commonly categorized as Nondeterministic Polynomial (NP)-hard combinatorial optimization problems and it remains as an endure problem to industrial practitioners and researchers. As part of real production scheduling, once one task is finished processing on a machine, transportation equipment such as mobile robot transports the completed task to the next machine. The problem of scheduling mobile robot in FMS pertains to the task allocation process for the robots, considering the transportation costs and the time spent to complete all operations. In recent years, Genetic Algorithm (GA) has been a remarkably effective search algorithm for solving a wide range of scheduling problems in a manner that achieves near-optimal solutions. This paper presents the metaheuristic techniques, specifically genetic algorithm, to address the NP-hard scheduling problem of two identical mobile robots in Job-Shop FMS environment. The algorithm is developed with the aim of finding feasible solutions to the integrated problem by minimizing the amount of time it takes to finish all tasks, commonly referred to as makespan. The performance of GA is evaluated with some numerical experiments which is executed via Matlab software. The scheduling results shows that the developed GA able to obtained the near-optimal solution of minimal makespan and converge within a short period of time.

An Efficient Closed-Form Formula for Evaluating r-Flip Moves in Quadratic Unconstrained Binary Optimization

Quadratic unconstrained binary optimization (QUBO) is a classic NP-hard problem with an enormous number of applications. Local search strategy (LSS) is one of the most fundamental algorithmic concepts and has been successfully applied to a wide range of hard combinatorial optimization problems. One LSS that has gained the attention of researchers is the r-flip (also known as r-Opt) strategy. Given a binary solution with n variables, the r-flip strategy “flips” r binary variables to obtain a new solution if the changes improve the objective function. The main purpose of this paper is to develop several results for the implementation of r-flip moves in QUBO, including a necessary and sufficient condition that when a 1-flip search reaches local optimality, the number of candidates for implementation of the r-flip moves can be reduced significantly. The results of the substantial computational experiments are reported to compare an r-flip strategy-embedded algorithm and a multiple start tabu search algorithm on a set of benchmark instances and three very-large-scale QUBO instances. The r-flip strategy implemented within the algorithm makes the algorithm very efficient, leading to very high-quality solutions within a short CPU time.

Research on Terminal Distance Index-Based Multi-Step Ant Colony Optimization for Mobile Robot Path Planning

Aiming at the path planning problem of mobile robot, when the traditional ant colony optimization (ACO) is simulated in grid model, it is assumed that ants can only move to adjacent nodes, that is, the step size is 1 and there are 8 movable directions. However, in practice, the moving direction of the ant is completely free. Therefore, an improved terminal distance index-based multi-step ant colony optimization (TDI-MSACO) for mobile robot path planning is proposed in this paper. A multi-step ant colony optimization (MSACO) is firstly used to improve the flexibility of ant’s movement and the path obtained by using MSACO is shorter and more in line with the actual situation, through the simulation of a large number of cases, it is determined that the optimal step size is 2 or 3. In addition, aiming at the problems of pheromone updating mechanism in MSACO, a concept of terminal distance index (TDI) is proposed to replace pheromone concentration and accelerate the convergence speed of the MSACO. In order to verify the effectiveness of the improved TDI-MSACO, the simulations are tested on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$20\times 20$</tex-math> </inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$30\times 30$</tex-math> </inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$50\times 50$</tex-math> </inline-formula> grid models and the results show that the improved TDI-MSACO has faster convergence speed and shorter path. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —the motivation of this paper comes from the need to develop a fast and efficient path planning method for practical applications such as material transport in shop floor, cleaning, monitoring of dangerous radioactive sites and military applications, and so on. The ant colony optimization (ACO), inspired from the foraging behavior of ant species, is a swarm intelligence algorithm for solving hard combinatorial optimization problems. The ACO is widely used in robot path planning areas because of its characteristics of positive feedback, distributed computation. However, it has the weaknesses of premature convergence and low search speed, which greatly hinder its application. To improve the performance of the ACO, an improved terminal distance index-based multi-step ant colony optimization (TDI-MSACO) for mobile robot path planning is proposed in this paper. Multi-step and terminal distance index are used to improve the flexibility of ant movement and accelerate the convergence speed, respectively. The effectiveness of the TDI-MSACO method for solving mobile path planning problems is proved by comparing the simulation results with those previously presented in the literature. Furthermore, the proposed method can be expanded to the dynamic environment containing multiple static and dynamic obstacles of many sizes and forms, it also can be used to solve the energy-efficient path planning problems. Meanwhile, it can be further modified and applied to the simultaneous scheduling of machines and transport robots in an actual flexible job shop environment.

Physics-Inspired Optimization in the QUBO Framework: Key Concepts and Approaches

Quantum computing promises to have a tremendous advantage over its classical counterpart for solving computationally hard problems, yet remains in a relatively early stage for practical applications owing to the limited capabilities of today’s quantum computers. The approach to the special purposes of quantum computers by exploiting the special-purpose physics-inspired or quantum-inspired computers is emerging as a novel alternative to its quantum counterpart in tackling hard problems in high-performance computing. Inspired by physics, the Ising machine — a type of special-purpose computer that implements or emulates physics or quantum effects of the Ising model to speed up finding solutions to optimization problems — has recently become an active research area in the field of combinatorial optimization. This paper is to address the key enabling software and hardware technology underlying physics-inspired optimization using Ising machines in the unified quadratic unconstrained binary optimization (QUBO) framework for modeling and solving computationally hard combinatorial optimization problems, and with an aim to shed some light on the challenges and opportunities associated with the ever-growing landscape of this novel high-performance computing.

Certified Dominance and Symmetry Breaking for Combinatorial Optimisation

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

Recognising permuted Demidenko matrices

We solve the recognition problem for permuted Demidenko matrices. This problem is relevant in the context of hard combinatorial optimization problems becoming tractable if the input is a Demidenko matrix. For example the TSP is polynomially solvable for Demidenko distance matrices. In the TSP context we look for a renumbering of the cities resulting in a Demidenko distance matrix, thus in a polynomially solvable case. We can find such a renumbering of n cities in O(n4) time, if it exists.

Learning to Break Symmetries for Efficient Optimization in Answer Set Programming

The ability to efficiently solve hard combinatorial optimization problems is a key prerequisite to various applications of declarative programming paradigms. Symmetries in solution candidates pose a significant challenge to modern optimization algorithms since the enumeration of such candidates might substantially reduce their performance. This paper proposes a novel approach using Inductive Logic Programming (ILP) to lift symmetry-breaking constraints for optimization problems modeled in Answer Set Programming (ASP). Given an ASP encoding with optimization statements and a set of small representative instances, our method augments ground ASP programs with auxiliary normal rules enabling the identification of symmetries using existing tools, like SBASS. Then, the obtained symmetries are lifted to first-order constraints with ILP. We prove the correctness of our method and evaluate it on real-world optimization problems from the domain of automated configuration. Our experiments show significant improvements of optimization performance due to the learned first-order constraints.

Designing Ising machines with higher order spin interactions and their application in solving combinatorial optimization

The Ising model provides a natural mapping for many computationally hard combinatorial optimization problems (COPs). Consequently, dynamical system-inspired computing models and hardware platforms that minimize the Ising Hamiltonian, have recently been proposed as a potential candidate for solving COPs, with the promise of significant performance benefit. However, prior work on designing dynamical systems as Ising machines has primarily considered quadratic interactions among the nodes. Dynamical systems and models considering higher order interactions among the Ising spins remain largely unexplored, particularly for applications in computing. Therefore, in this work, we propose Ising spin-based dynamical systems that consider higher order (> 2) interactions among the Ising spins, which subsequently, enables us to develop computational models to directly solve many COPs that entail such higher order interactions (i.e., COPs on hypergraphs). Specifically, we demonstrate our approach by developing dynamical systems to compute the solution for the Boolean NAE-K-SAT (K ≥ 4) problem as well as solve the Max-K-Cut of a hypergraph. Our work advances the potential of the physics-inspired ‘toolbox’ for solving COPs.

Cutting and packing problems under uncertainty: literature review and classification framework

AbstractCutting and packing problems are hard combinatorial optimization problems that arise in several manufacturing and process industries or in their supply chains. The solution of these problems is not only a scientific challenge but also has a large economic impact, as it contributes to the reduction of one of the major cost factors for many production sectors, namely raw materials, together with a positive environmental impact. The explicit consideration of uncertainty when solving cutting and packing problems with optimization techniques is crucial for a wider adoption of research results by companies. However, current research has paid little attention to the role of uncertainty in these problems. In this paper, we review the existing literature on uncertainty in cutting and packing problems, propose a classification framework, and highlight the many research gaps and opportunities for scientific contributions.

Binary arithmetic optimization algorithm for feature selection.

Feature selection, widely used in data preprocessing, is a challenging problem as it involves hard combinatorial optimization. So far some meta-heuristic algorithms have shown effectiveness in solving hard combinatorial optimization problems. As the arithmetic optimization algorithm only performs well in dealing with continuous optimization problems, multiple binary arithmetic optimization algorithms (BAOAs) utilizing different strategies are proposed to perform feature selection. First, six algorithms are formed based on six different transfer functions by converting the continuous search space to the discrete search space. Second, in order to enhance the speed of searching and the ability of escaping from the local optima, six other algorithms are further developed by integrating the transfer functions and Lévy flight. Based on 20 common University of California Irvine (UCI) datasets, the performance of our proposed algorithms in feature selection is evaluated, and the results demonstrate that BAOA_S1LF is the most superior among all the proposed algorithms. Moreover, the performance of BAOA_S1LF is compared with other meta-heuristic algorithms on 26 UCI datasets, and the corresponding results show the superiority of BAOA_S1LF in feature selection. Source codes of BAOA_S1LF are publicly available at: https://www.mathworks.com/matlabcentral/fileexchange/124545-binary-arithmetic-optimization-algorithm.

Identifying the cardinality-constrained critical nodes with a hybrid evolutionary algorithm

Identifying the critical nodes in a network is crucial for understanding its characteristics, controlling its structure, and determining its functionality. Cardinality-constrained CNP (CC-CNP) is a nondeterministic polynomial-time (NP)-hard combinatorial optimization problem that refers to minimizing a set of nodes such that after deletion, the size of the largest connected component in the residual subgraph is smaller than a prescribed value. CC-CNP is applicable to a variety of fields, such as epidemic and infectious disease control, electric power network construction and maintenance, and traffic network control.In this work, we present a multistage local search (MSLS) algorithm for generating high-quality initial solutions for CC-CNP, where two strategies, circular node deletion and node change and the tabu search-based first in, first out (FIFO) principle, are utilized to prevent search detours. Then, a population-based strategy is incorporated, resulting in a genetic algorithm-based multistage local search algorithm (GAMSLS) that adopts a genetic algorithm framework, refines the initial solutions in the crossover process, and utilizes a new population update strategy to ensure the diversity and individual quality of the population. The proposed algorithm is evaluated on 75 network instances and is shown to outperform state-of-the-art algorithms for CC-CNP.

MIP-based heuristics for combinatorial design of reconfigurable rotary transfer machines for production of multiple parts

This paper deals with a problem of the optimal configuration of a rotary transfer machine with turrets for machining multiple parts. This is a hard combinatorial optimization problem appearing at the preliminary machine design stage. Such machines are multi-positional, i.e. parts are sequentially machined on several working positions. At each working position, several machining modules (spindle heads) can be installed to process the tasks assigned to this position. Machining modules are activated sequentially or simultaneously. Sequential activation is realized by the use of turrets. Simultaneous activation is possible if machining modules are related to the different sides of the part, and if they can work in parallel. There are horizontal and vertical spindle heads, and turrets to access different sides of the parts on a working position. At the preliminary design stage, the following decisions must be made: the choice of orientations of the parts on the rotary table; the partitioning of the given set of tasks into positions and their assignment to machining modules (selection or design of machining modules to use), and the choice of cutting modes for each spindle head and turret. The objective is to minimize the total cost of equipment used. The number of possible solutions for this combinatorial design problem increases exponentially with the number of part types to be produced, and this represents a computational burden for decision-makers (usually process engineers). In this paper, in order to help decision makers deal efficiently with the manufacturing of multiple batches of parts, we develop a powerful heuristic framework which can be used in real life industrial cases. We test the developed methodology on the real-life cases provided by one of our industrial partners and demonstrate its efficiency. The proposed model and algorithms allow to minimize the cost of designed machines.