Applications In Combinatorial Optimization Research Articles

In-Materia Annealing and Combinatorial Optimization Based on Vertical Memristive Array.

Due to its area and energy efficiency, a memristive crossbar array (CBA) has been extensively studied for various combinatorial optimization applications, from network problems to circuit design. However, conventional approaches include heavily burdening software fine-tuning for the annealing process. Instead, this study introduces the "in-materia annealing" method, where the inter-layer interference of vertically stacked memristive CBA is utilized as an annealing method. When mapping combinatorial optimization problems into the configuration layer of the CBA, exponentially decaying annealing profiles are generated in nearby noise layers. Moreover, in-materia annealing profiles can be controlled by changing compliance current, read voltage, and read pulse width. Therefore, the annealing profiles can be arbitrarily controlled and generated individually for each cell, providing rich noise sources to solve the problem efficiently. Consequently, the experimental and simulation of Max-Cut and weighted Max-Cut problems achieve notable results with the minimum software burden.

A novel parallel heuristic method to design a sustainable medical waste management system

Efficient management of waste generated in healthcare systems is crucial to minimize its environmental impact and ensure public health. Sustainable medical waste management (MWM) systems require careful network design, which can be achieved through efficient optimization techniques. This work develops a mixed-integer linear programming (MILP) to formulate the problem, a two-step MILP (TSMILP) to generate quality lower bounds, and a novel parallel heuristic algorithm to configure a sustainable waste management system including waste generation centers (WGCs), waste treatment centers (WTCs), waste recycling centers (WRCs), waste disposal centers (WDCs) and waste incineration centers (WICs). Such a hybrid methodology has not been yet offered in the literature wherein the aim is to address strategic (establishment of facilities), tactical (employment of transportation system), and operational decisions (transportation planning) optimally in large networks. As reflected in the literature, there is a huge gap in efficiency and application of combinatorial optimization, and parallel computing in sustainable MWM systems, where the suggested MILPs' solvers are not technically capable of discovering quality solutions in reasonable runtimes on large-sized instances. Thus, we suggest a novel heuristic equipped with parallel computing to share the complexity of the problem, with all the CPU cores to shorten runtime. Comparing the results generated by the parallel heuristic with those of the sequential heuristic, the MILP, and the TSMILP on three sets of benchmark instances using Nemenyi's post-hoc procedure for Friedman's test, it is inferred that the parallel heuristic is so effective in coping with the problem, and produces high-quality solutions, especially on the large-sized set. Finally, sensitivity analysis is adopted to analyze the effects of parameters on the objective values and provide useful managerial insights.

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

AbstractIn this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

A Twin Learning Framework for Traveling Salesman Problem Based on Autoencoder, Graph Filter, and Transfer Learning

Efficient solvers for traveling salesman problem (TSP) have great significance in the field of consumer electronic systems and devices. Existing studies require independent and repetitive runs for similar TSPs. To utilize useful knowledge buried in the twin TSP which is similar to the target TSP, a twin learning framework based on task matching and mapping strategy is proposed. We use an autoencoder to extract the feature vectors of the historical tasks and the target task to find the twin task. If not found, construct a twin task based on a graph-filter. Further, we get the solution of the target task with the optimized twin task by learning a mapping matrix from the twin task to the target task. Finally, we incorporate local search algorithms into the twin learning framework, which called strategy mapping solver (SMS) to further improve the quality of the solution. The efficacy of the SMS is evaluated through comprehensive empirical studies with commonly used TSP benchmarks, against meta-heuristic algorithms and a learning improvement heuristics algorithm, demonstrating its effectiveness for TSP. Moreover, a real-world combinatorial optimization application, laser engraving path planning, is presented to further confirm the efficacy of SMS.

Improved cooperative Ant Colony Optimization for the solution of binary combinatorial optimization applications

AbstractBinary combinatorial optimization plays a crucial role in various scientific and engineering fields. While deterministic approaches have traditionally been used to solve these problems, stochastic methods, particularly metaheuristics, have gained popularity in recent years for efficiently handling large problem instances. Ant Colony Optimization (ACO) is among the most successful metaheuristics and is frequently employed in non‐binary combinatorial problems due to its adaptability. Although for binary combinatorial problems ACO can suffer from issues such as rapid convergence to local minima, its eminently parallel structure means that it can be exploited to solve large and complex problems also in this field. In order to provide a versatile ACO implementation that achieves competitive results across a wide range of binary combinatorial optimization problems, we introduce a parallel multicolony strategy with an improved cooperation scheme to maintain search diversity. We evaluate our proposal (Binary Parallel Cooperative ACO, BiPCACO) using a comprehensive benchmark framework, showcasing its performance and, most importantly, its flexibility as a successful all‐purpose solver for binary combinatorial problems.

К ВОПРОСУ О ФОРМАЛИЗАЦИИ ЗАДАЧИ РАНЖИРОВАНИЯ ТРЕБОВАНИЙ ИНФОРМАЦИОННОЙ БЕЗОПАСНОСТИ

The article is devoted to the issue of requirements fulfillment aimed at ensuring information security in an organization. The main contradiction of the subject area concerned is pointed out, which consists in the presence of a huge number of different variants of requirements fulfillment in the absence of a possibility to choose their correct and optimal order. The task of requirements ranking is set and the idea of the proposed solution is described in the form of seven provisions aimed at coordinated recording of heterogeneous requirements in a single notation, and an intuitive scheme of the idea is synthesized (with all seven provisions indicated on it).
 To represent the idea, the following entities are introduced: an object-organization and its elements to which requirements are imposed; generalized conditions for satisfying requirements that do not depend on the specifics of the organization; variations of sets of conditions that take into account a particular organization; basic conditions that check the presence/absence of elements 
 of the object and the values of their parameters; algorithms of activities in the organization to satisfy the conditions; priorities of requirements and resources needed by the algorithms. It is concluded that such formalization will lead organically to the algorithmic solution of the ranking problem and, eventually, to automation.
 The most suitable automated ways of solving the problem of ranking information security requirements – algorithmic application of combinatorial optimization and machine learning methods – are specified. Their high efficiency in comparison with «manual» methods used in modern information protection practice is predicted.
 The novelty, theoretical and practical significance of the obtained results are noted, as well as the prospect of further research – the construction of an analytical model of requirements fulfillment, which could be the basis of an appropriate method, followed by its program implementation and conducting of necessary experiments.

New advances for quantum‐inspired optimization

AbstractAdvances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.

The applications of hybrid approach combining exact method and evolutionary algorithm in combinatorial optimization

Abstract Combinatorial optimization problems have very important applications in information technology, transportation, economics, management, network communication, and other fields. Since the problem size in real-scenario application is in large-scale, the demand for real-time and efficient solving approaches increases rapidly. The traditional exact methods guarantee the optimality of the final solution, but these methods can hardly solve the problem in acceptable time due to extremely high computational costs. Heuristic approaches can find feasible solutions in a limited time, while these approaches cannot meet the demand of solution quality. In recent years, hybrid algorithms based on exact methods and heuristic algorithms show outstanding performance in solving large-scale combinatorial optimization problems. The hybridization not only overcomes the shortcomings from single algorithm but also fully utilizes the search ability for population-based approaches as well as the interpretability in exact methods, which promotes the application of combinatorial optimization in real-world problems. This paper reviews existing studies on hybrid algorithms combining exact method and evolutionary computation, summarizes the characteristics of the existing algorithms, and directs the future research.

Distributed Synchronous and Asynchronous Algorithms for Semidefinite Programming With Diagonal Constraints

This article develops distributed synchronous and asynchronous algorithms for the large-scale semidefinite programming with diagonal constraints, which has wide applications in combinatorial optimization, image processing, and community detection. The information of the semidefinite programming is allocated to multiple interconnected agents such that each agent aims to find a solution by communicating to its neighbors. Based on the low-rank property of solutions and the Burer–Monteiro factorization, we transform the original problem into a distributed optimization problem over unit spheres to reduce variable dimensions and ensure positive semidefiniteness without involving semidefinite projections, which are computationally expensive. For the distributed optimization problem, we propose distributed synchronous and asynchronous algorithms, both of which reduce computational burden and storage space compared with existing centralized algorithms. Specifically, the distributed synchronous algorithm almost surely escapes strict saddle points and converges to the set of optimal solutions to the optimization problem. In addition, the proposed distributed asynchronous algorithm allows communication delays and converges to critical points to the optimization problem under mild conditions. By applying the proposed algorithms to image segmentation, we illustrate the efficiency and convergence performance of the two proposed algorithms.

Explicit Evolutionary Multitasking for Combinatorial Optimization: A Case Study on Capacitated Vehicle Routing Problem.

Recently, evolutionary multitasking (EMT) has been proposed in the field of evolutionary computation as a new search paradigm, for solving multiple optimization tasks simultaneously. By sharing useful traits found along the evolutionary search process across different optimization tasks, the optimization performance on each task could be enhanced. The autoencoding-based EMT is a recently proposed EMT algorithm. In contrast to most existing EMT algorithms, which conduct knowledge transfer across tasks implicitly via crossover, it intends to perform knowledge transfer explicitly among tasks in the form of task solutions, which enables the employment of task-specific search mechanisms for different optimization tasks in EMT. However, the autoencoding-based explicit EMT can only work on continuous optimization problems. It will fail on combinatorial optimization problems, which widely exist in real-world applications, such as scheduling problem, routing problem, and assignment problem. To the best of our knowledge, there is no existing effort working on explicit EMT for combinatorial optimization problems. Taking this cue, in this article, we thus embark on a study toward explicit EMT for combinatorial optimization. In particular, by using vehicle routing as an illustrative combinatorial optimization problem, the proposed explicit EMT algorithm (EEMTA) mainly contains a weighted l1 -norm-regularized learning process for capturing the transfer mapping, and a solution-based knowledge transfer process across vehicle routing problems (VRPs). To evaluate the efficacy of the proposed EEMTA, comprehensive empirical studies have been conducted with the commonly used vehicle routing benchmarks in multitasking environment, against both the state-of-the-art EMT algorithm and the traditional single-task evolutionary solvers. Finally, a real-world combinatorial optimization application, that is, the package delivery problem (PDP), is also presented to further confirm the efficacy of the proposed algorithm.

Data Association via Set Packing for Computer Vision Applications

Significant progress has been made in the field of computer vision because of the development of supervised machine learning algorithms, which efficiently extract information from high-dimensional data such as images and videos. Such techniques are particularly effective at recognizing the presence or absence of entities in the domains where labeled data are abundant. However, supervised learning is not sufficient in applications where one needs to annotate each unique entity in crowded scenes respecting known domain-specific structures of those entities. This problem, known as data association, provides fertile ground for the application of combinatorial optimization. In this review paper, we present a unified framework based on column generation for some computer vision applications, namely multiperson tracking, multiperson pose estimation, and multicell segmentation, which can be formulated as set packing problems with a massive number of variables. To solve them, column generation algorithms are applied to circumvent the need to enumerate all variables explicitly. To enhance the solution process, we provide a general approach for applying subset-row inequalities to tighten the formulations and introduce novel dual-optimal inequalities to reduce the dual search space. The proposed algorithms and their enhancements are successfully applied to solve the three aforementioned computer vision problems and achieve superior performance over benchmark approaches. The common framework presented allows us to leverage operations research methodologies to efficiently tackle computer vision problems.

Advancing CMOS-Type Ising Arithmetic Unit into the Domain of Real-World Applications

Solving combinatorial optimization problems is a great challenge for Von Neumann-architecture computing. Although the Ising model could provide promising solutions for such problems, existing Ising chips, including superconductive, optical and CMOS-type circuit implementation, cannot meet the precision requirement of real-world combinatorial optimization applications. To facilitate the support for real-world applications, we propose three improvements over existing CMOS-type Ising chips: suitable narrow bit width memory cells with approximate multiply-adders, double random sources flipping method with cross random number generators and shared circuit design between adjacent spin nodes. With above improvements, we achieve high precision as well as maintaining the low cost characteristic of CMOS-type Ising chips. When searching the ground state of Ising models, our CMOS-type Ising chip can improve the precision to more than 99 percent over existing ones with about 93 percent precision. Moreover, its hardware cost is only 32 percent of the common implementation to achieve the same high precision. Specially, we have applied our Ising chip in image segmentation applications, a typical real-world application. The results show that, to find a segmentation with similar quality, our CMOS-type Ising chip can speed up the segmenting processing by 1900× with only 0.017‰ energy consumption compared with approximate algorithms operating on conventional computers.

Submodular functions: from discrete to continuous domains

Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains, where it corresponds essentially to cross second-derivatives being nonpositive. In this paper, we show that most results relating submodularity and convexity for set-functions can be extended to all submodular functions. In particular, (a) we naturally define a continuous extension in a set of probability measures, (b) show that the extension is convex if and only if the original function is submodular, (c) prove that the problem of minimizing a submodular function is equivalent to a typically non-smooth convex optimization problem, and (d) propose another convex optimization problem with better computational properties (e.g., a smooth dual problem). Most of these extensions from the set-function situation are obtained by drawing links with the theory of multi-marginal optimal transport, which provides also a new interpretation of existing results for set-functions. We then provide practical algorithms to minimize generic submodular functions on discrete domains, with associated convergence rates.

Application of combinatorial optimization of sensor electrode plate in three dimensional ECT system in real-time imaging

ECT technology has the advantages of simple structure, fast response, non-invasion, no pollution, and no radiation hazard and so on. Because of these advantages of ECT technology, this technology has been widely used for various applications such as flow measurement in chemical industry and petroleum; visualize the tooth surface using dental ECT, real-time medical imaging, revision total hip replacement etc. The research of ECT technology can be divided into two-dimensional ECT and three-dimensional ECT technology, and the current three-dimensional ECT technology has been widely studied.

Maximizing a Class of Utility Functions Over the Vertices of a Polytope

Given a polytope X, a monotone concave univariate function g, and two vectors c and d, we study the discrete optimization problem of finding a vertex of X that maximizes the utility function c’x + g(d’x). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models, in multinomial logit models and in robust optimization. We show that the problem is 𝒩𝒫-hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. We also propose a 1.25-approximation algorithm for a class of minimization problems in which the maximization of the utility function appears as a subproblem. Although the worst-case bounds are tight, computational experiments indicate that the suggested approach finds solutions within 1%–2% optimality gap for most of the instances, and can be considerably faster than the existing alternatives.

Robust efficiency measures for linear knapsack problem variants

Numerous combinatorial optimization applications, such as the mobile retailer product mix problem, utilize the multidemand, multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value. The purpose of this paper is to demonstrate that core variables and efficiency measures, concepts used in the most efficient solvers to-date for binary knapsack problems and some of its variants, can be extended to the multidemand, multidimensional knapsack problem variant. Specifically, new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated. The contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity. The applicability of these new measures is demonstrated through the development of three heuristic procedures: a Fixed-Core heuristic, a Genetic Algorithm, and a Kernel Search heuristic. The results from these tests are compared with the results from a commercial solver and an existing heuristic. The findings from these tests demonstrate that the Fixed-Core and Kernel Search heuristics developed for this paper are the most efficient solvers to-date for hard multidemand, multidimensional knapsack problems.

Test case minimization approach using fault detection and combinatorial optimization techniques for configuration-aware structural testing

This paper presents a technique to minimize the number of test cases in configuration-aware structural testing. Combinatorial optimization is used first to generate an optimized test suite by sampling the input configuration. Second, for further optimization, the generated test suite is filtered based on an adaptive mechanism by using a mutation testing technique. The initialized test suite is optimized using cuckoo search (CS) along with combinatorial approach, and mutation testing is used to seed different faults to the software-under-test, as well as to filter the test cases based on the detected faults. To measure the effectiveness of the technique, an empirical study is conducted on a software system. The technique proves its effectiveness through the conducted case study. The paper also shows the application of combinatorial optimization and CS to the software testing.

An efficient strategy for covering array construction with fuzzy logic-based adaptive swarm optimization for software testing use

Recent research activities have demonstrated the effective application of combinatorial optimization in different areas, especially in software testing. Covering array (CA) has been introduced as a representation of the combinations in one complete set. CAλ(N; t, k, v) is an N × k array in which each t-tuple for an N × t sub array occurs at least λ times, where t is the combination strength, k is the number of components (factors), and v is the number of symbols for each component (levels). Generating an optimized covering array for a specific number of k and v is difficult because the problem is a non-deterministic polynomial-time hard computational one. To address this issue, many relevant strategies have been developed, including stochastic population-based algorithms. This paper presents a new and effective approach for constructing efficient covering arrays through fuzzy-based, adaptive particle swarm optimization (PSO). With this approach, efficient covering arrays have been constructed and the performance of PSO has been improved for this type of application. To measure the effectiveness of the technique, an empirical study is conducted on a software system. The technique proves its effectiveness through the conducted case study.

Solving multi-objective job shop problem using nature-based algorithms: new Pareto approximation features

In this paper the job shop scheduling problem (JSP) with minimizing two criteria simultaneously is considered. JSP is frequently used model in real world applications of combinatorial optimization. Multi-objective job shop problems (MOJSP) were rarely studied. We implement and compare two multi-agent nature-based methods, namely ant colony optimization (ACO) and genetic algorithm (GA) for MOJSP. Both of those methods employ certain technique, taken from the multi-criteria decision analysis in order to establish ranking of solutions. ACO and GA differ in a method of keeping information about previously found solutions and their quality, which affects the course of the search. In result, new features of Pareto approximations provided by said algorithms are observed: aside from the slight superiority of the ACO method the Pareto frontier approximations provided by both methods are disjoint sets. Thus, both methods can be used to search mutually exclusive areas of the Pareto frontier.

Introduction to special xQx issue

The Unconstrained Binary Quadratic Programming (UBQP) problem is notable for encompassing a remarkable range of applications in combinatorial optimization. As observed in Kochenberger et al. (2013), classes of problems that can be formally expressed using UBQP formulations include: Quadratic Assignment Problems, Capital Budgeting Problems, Multiple Knapsack Problems, Task Allocation Problems (distributed computer systems), Maximum Diversity Problems, P-Median Problems, Asymmetric Assignment Problems, Symmetric Assignment Problems, Side Constrained Assignment Problems, Quadratic Knapsack Problems, Constraint Satisfaction Problems (CSPs), Set Partitioning Problems, Fixed Charge Warehouse Location Problems, Maximum Clique Problems, Maximum Independent Set Problems, Maximum Cut Problems, Graph Coloring Problems, Graph Partitioning Problems, Number Partitioning Problems, and Linear Ordering Problems. Even more remarkable is the fact that, once given a UBQP formulation, these problems can be solved by a UBQP method which is not specialized to exploit the problem domain of any individual class of problems, to yield solutions whose quality in many cases rivals or even surpasses the quality of the solutions produced by the best specialized methods, while achieving this outcome with an efficiency that likewise rivals or surpasses the efficiency of leading specialized methods. Moreover, these outcomes typically dominate those produced by current state-of-the-art commercial solvers for mixed integer linear and quadratic optimization, sometimes consuming two orders of magnitude less solution time to yield solutions superior to those of competing approaches. In other cases, where specialized methods edge out general UBQP methods for certain classes of problems, the UBQP formulation often still