Editorial for the Special Issue on Combinatorial Optimization Problems

First paragraph: In combinatorial optimization, the goal is to find an optimal solution, according to some objective function, from a discrete search space. These problems arise widely in industry and academia and, unfortunately, many of them are NP-hard and no polynomial time algorithm can guarantee their solution to a certified optimality unless. Therefore, in the last decades researchers have investigated the use of stochastic search algorithms to find near optimal solutions to these problems. In particular, great research efforts have been devoted to the development and application of metaheuristic algorithms to solve combinatorial optimization problems

A Survey on Influence Maximization: From an ML-Based Combinatorial Optimization

Influence Maximization (IM) is a classical combinatorial optimization problem, which can be widely used in mobile networks, social computing, and recommendation systems. It aims at selecting a small number of users such that maximizing the influence spread across the online social network. Because of its potential commercial and academic value, there are a lot of researchers focusing on studying the IM problem from different perspectives. The main challenge comes from the NP-hardness of the IM problem and \#P-hardness of estimating the influence spread, thus traditional algorithms for overcoming them can be categorized into two classes: heuristic algorithms and approximation algorithms. However, there is no theoretical guarantee for heuristic algorithms, and the theoretical design is close to the limit. Therefore, it is almost impossible to further optimize and improve their performance. With the rapid development of artificial intelligence, the technology based on Machine Learning (ML) has achieved remarkable achievements in many fields. In view of this, in recent years, a number of new methods have emerged to solve combinatorial optimization problems by using ML-based techniques. These methods have the advantages of fast solving speed and strong generalization ability to unknown graphs, which provide a brand-new direction for solving combinatorial optimization problems. Therefore, we abandon the traditional algorithms based on iterative search and review the recent development of ML-based methods, especially Deep Reinforcement Learning, to solve the IM problem and other variants in social networks. We focus on summarizing the relevant background knowledge, basic principles, common methods, and applied research. Finally, the challenges that need to be solved urgently in future IM research are pointed out.Comment: 45 page

Algorithmic and Statistical Perspectives on Large-Scale Data Analysis

In recent years, ideas from statistics and scientific computing have begun to interact in increasingly sophisticated and fruitful ways with ideas from computer science and the theory of algorithms to aid in the development of improved worst-case algorithms that are useful for large-scale scientific and Internet data analysis problems. In this chapter, I will describe two recent examples---one having to do with selecting good columns or features from a (DNA Single Nucleotide Polymorphism) data matrix, and the other having to do with selecting good clusters or communities from a data graph (representing a social or information network)---that drew on ideas from both areas and that may serve as a model for exploiting complementary algorithmic and statistical perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors, "Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201

A discrete simulated kalman filter optimizer for combinatorial optimization problems

Combinatorial optimization problems are ubiquitous in many fields, including healthcare, economics, engineering, manufacturing, and others. A solution to a combinatorial optimization problem is frequently expressed in terms of a permutation, arrangement, or combination of elements. Due to the practical significance of this problem in real-world issues, numerous algorithms have been proposed to solve it. These algorithms specifically refer to those that operate in discrete search space, often known as combinatorial algorithms. Another type of algorithm is called numerical algorithms. These algorithms were built specifically to address numerical optimization problems. In the last few decades, significant research effort has been spent on the development of numerical algorithms, particularly for solving combinatorial problems. An example of a numerical algorithm is the simulated Kalman filter (SKF). Various method has been introduced as an extension of a numerical algorithm to adapt it to a discrete search space. There are currently three extensions to the SKF, resulting in three combinatorial algorithms: the binary SKF (BSKF), the distance evaluated SKF (DESKF), and the angle modulated SKF (AMSKF). However, these extensions may result in increased execution times for the algorithm. In this research, a new combinatorial algorithm named discrete simulated Kalman filter optimizer (DSKFO) is proposed to solve combinatorial optimization problem. This new algorithm is originated by the concept of the simulated Kalman filter (SKF). Due to the limitation of the SKF algorithm which only able to operate in continuous search space, the proposed algorithm makes use of a new interpretation that incorporates mutation and Hamming distance, allowing the proposed algorithm to function in discrete search space. In this research, three combinatorial problems namely the travelling salesman problem (TSP), assembly sequence planning (ASP), and the hole drilling proble are used to evaluate the proposed algorithm. Two types of analysis are used to evaluate the proposed algorithm. First, the DSKFO algorithm is used to solve the travelling salesman problem (TSP), and then the algorithm's execution time is measured. Existing SKF methods are then compared to the findings of the DSKFO algorithm. DSKFO performs the fastest, requiring just 13 seconds to solve a small TSP instance such as eil51, whereas DESKF, AMSKF, BSKF, and SEDESKF require around 36, 42, 34, and 14 seconds, respectively. To solve larger TSP instance such as rl1889, DSKFO requires 139 seconds to execute a single run, whereas DESKF, AMSKF, BSKF, and SEDESKF require around 1587, 1590, 2418, and 208 seconds, respectively. For the second analysis, the performance of the proposed method is evaluated using three combinatorial problems: the travelling salesman problem (TSP), the assembly sequence planning (ASP), and the hole drilling problem. The results are compared to four previously published combinatorial SKFs: the BSKF, the AMSKF, the DESKF, and the SEDESKF. The DSKFO may be considered the best algorithm for solving the TSP and hole drilling problem, as it has the highest number of best performances. For solving the ASP, the DSKFO ranked third, while the AMSKF came in first, followed by the DESKF in second

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Оптимізаційні задачі на переставленнях: метод комбінаторного відсікання з використанням алгоритму Кармаркара

Combinatorial optimization problems have been intensively researched in recent years, which lead to the development of new approaches and methods to solve these tasks. Important is the development of algorithms for solving combinatorial optimization. Method for solving combinatorial cut-off method of linear programming problems using Karmarkar’s algorithm is proposing in this paper. The article describes the method of combinatorial cut-off based on the Karmarkar algorithm. Formulated and proved a theorem about the finiteness of the proposed method.Останнім часом інтенсивно досліджуються задачі комбінаторної оптимізації, що призводить до розробки нових підходів та методів до їх розв’язування. Актуальною є розробка поліноміальних алгоритмів для розв’язування задач комбінаторної оптимізації. У цьому дослідженні запропоновано використовувати алгоритм Кармаркара у методі комбінаторного відсікання для розв’язування допоміжних задач лінійного програмування. Викладено метод комбінаторного відсікання на основі алгоритму Кармаркара. Сформульовано та доведено теорему про скінченність запропонованого методу

Algorithms for Large Orienteering Problems

In this thesis, we have developed algorithms to solve large-scale Orienteering Problems. The Orienteering Problem is a combinatorial optimization problem were given a weighted complete graph with vertex profits and a maximum distance constraint, the goal is to find the simple cycle which maximizes the sum of the profits of the visited vertices. To solve the Orienteering Problem, we have developed an evolutionary algorithm and an Branch-and-Cut algorithm. One of the key characteristics of the evolutionary algorithm is to work with unfeasible solutions. From the point of view of genetic operators, the main contribution has been the development of the Edge Recombination Crossover for the Orienteering Problem, which in a wider context it is also valid for any cycle problem. Another contribution has been the developed local search to handle large problems. The Branch-and-Cut algorithm includes new contributions in the separation algorithms of inequalities stemming from the cycle problem, in the separation loop, in the variables pricing, and in the calculation of the lower and upper bounds of the problem. At the same time, we have generalized for cycle problems the support graph shrinking techniques and procedures to speed up the exact separation algorithms for subcycle elimination constraints. The experiments carried out in large-sized instances, up to 7393 nodes, show that both algorithms achieve outstanding results, both in terms of the quality of solutions and in terms of the execution time.BERC.2014-2017 SEV-2013-0323 PID2019-104933GB-I00 MTM2015-65317-

Developments in linear and integer programming

In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies

A reusable iterative optimization software library to solve combinatorial problems with approximate reasoning

Real world combinatorial optimization problems such as scheduling are typically too complex to solve with exact methods. Additionally, the problems often have to observe vaguely specified constraints of different importance, the available data may be uncertain, and compromises between antagonistic criteria may be necessary. We present a combination of approximate reasoning based constraints and iterative optimization based heuristics that help to model and solve such problems in a framework of C++ software libraries called StarFLIP++. While initially developed to schedule continuous caster units in steel plants, we present in this paper results from reusing the library components in a shift scheduling system for the workforce of an industrial production plant.Comment: 33 pages, 9 figures; for a project overview see http://www.dbai.tuwien.ac.at/proj/StarFLIP