Hard Combinatorial Optimization Problems Research Articles

Regularized greedy column subset selection

The Column Subset Selection Problem is a hard combinatorial optimization problem that provides a natural framework for unsupervised feature selection, and there exist efficient algorithms that provide good approximations. The drawback of the problem formulation is that it incorporates no form of regularization, and is therefore very sensitive to noise when presented with scarce data. In this paper we propose a regularized formulation of this problem, and derive a correct greedy algorithm that is similar in efficiency to existing greedy methods for the unregularized problem. We study its adequacy for feature selection and propose suitable formulations. Additionally, we derive a lower bound for the error of the proposed problems. Through various numerical experiments on real and synthetic data, we demonstrate the significantly increased robustness and stability of our method, as well as the improved conditioning of its output, all while remaining efficient for practical use.

New algorithmic framework for conditional value at risk: Application to stochastic fixed-charge transportation

This paper introduces a new algorithmic scheme for two-stage stochastic mixed-integer programming assuming a risk averse decision maker. The focus is the minimization of the conditional value at risk for a hard combinatorial optimization problem. Some properties of a mixed-integer non-linear programming formulation for conditional value at risk are studied as well as their algorithmic implications. This yields to a procedure for obtaining lower and upper bounds on the optimal value of the problem that may lead to an optimal solution. The new developments are applied to a fixed-charge transportation problem with stochastic demand, and they are computationally tested. The corresponding results are thoroughly presented and discussed.

Multi-objective ant colony optimization algorithm based on decomposition for community detection in complex networks

Community detection aims to identify topological structures and discover patterns in complex networks, which presents an important problem of great significance. The problem can be modeled as an NP hard combinatorial optimization problem, to which multi-objective optimization has been applied, addressing the common resolution limitation problem in modularity-based optimization. In the literature, ant colony optimization (ACO) algorithm, however, has been only applied to community detection with single objective. This is due to the main difficulties in defining and updating the pheromone matrices, constructing the transition probability model, and tuning the parameters. To address these issues, a multi-objective ACO algorithm based on decomposition (MOACO/D-Net) is proposed in this paper, minimizing negative ratio association and ratio cut simultaneously in community detection. MOACO/D-Net decomposes the community detection multi-objective optimization problem into several subproblems, and each one corresponds to one ant in the ant colony. Furthermore, the ant colony is partitioned into groups, and ants in the same group share a common pheromone matrix with information learned from high-quality solutions. The pheromone matrix of each group is updated based on updated nondominated solutions in this group. New solutions are constructed by the ants in each group using a proposed transition probability model, and each of them is then improved by an improvement operator based on the definition of strong community. After improvement, all the solutions are compared with the solutions in the external archive and the nondominated ones are added to the external archive. Finally each ant updates its current solution based on a better neighbor, which may belong to an adjacent group. The resulting final external archive consists of nondominated solutions, and each one corresponds to a different partition of the network. Systematic experiments on LFR benchmark networks and eight real-world networks demonstrate the effectiveness and robustness of the proposed algorithm. The ranges of proper values for each parameter are also analyzed, addressing the key issue of parameter tuning in ACO algorithms based on a large number of tests conducted.

A new hybrid genetic algorithm for the maximally diverse grouping problem

This paper presents a new hybrid approach ( $$\mathcal {N}$$ SGGA) combining steady-state grouping genetic algorithm with a local search for the maximally diverse grouping problem (MDGP) related to equal group-size. The MDGP is a well-known $$\mathcal {NP}$$ -Hard combinatorial optimization problem and finds numerous applications in real world. $$\mathcal {N}$$ SGGA employs particularly (a) crossover operator (b) the effective way of utilization of local search and (c) the additional replacement strategy, making it different from the existing genetic algorithm for the MDGP. On a set of large benchmark instances, $$\mathcal {N}$$ SGGA is competitive in comparison to the existing best-known approaches in the literature and performs particularly well on large-size instances. Some important ingredients of $$\mathcal {N}$$ SGGA that shed some light on the adequacy of $$\mathcal {N}$$ SGGA are analyzed.

A new leader guided optimization for the flexible job shop problem

The FJSP is an extension of the classical job shop problem which has been proven to be among the hardest combinatorial optimization problems, by allowing an operation to be operated on more than one machine from a machine set, with possibility of variable performances. In this work, we have designed a co-evolutionary algorithm that applies adaptively multiple crossover and mutation operators. In the evolution process, all new generated individuals are improved by local search. Combined with a new leader tree guided optimization search, the hybrid algorithm has discovered 2 new optimal solutions for instances of Hurink et al. (Oper Res Spektrum 15(4):205–215, 1994). In general, the outcomes of simulation results and comparisons demonstrate comparable results. The leader guided optimization has shown its effectiveness for minimizing the makespan in a FJSP, but it is not limited to this environment.

A new hybrid gravitational particle swarm optimisation-ACO with local search mechanism, PSOGSA-ACO-Ls for TSP

The travelling salesman problem (TSP) is a hard combinatorial optimisation problem and a popular benchmarking problem at the same time. The TSP has also a number of practical real-world and industrial applications, such as routing in internet of things, IoT, networks, path planning in robotics and many others. In this paper, a new hybrid algorithm for the TSP is proposed; it combines gravitational particle swarm optimisation (PSOGSA) and ACO, and is called ant supervised by gravitational particle swarm optimisation with a local search, PSOGSA-ACO-LS. PSOGSA is used to optimise ACO settings while a local search mechanism, 2-Opt is employed by ACO to ameliorate its local solutions. The proposed method is evaluated using a set a test benches from the TSPLib database including: eil51, berlin52, st70, eil76, rat99, eil101, kroA100, and kroA200. Experimental results show that ACO-GPSO-LS is able to solve the set of TSP instances listed below including the large TSP data sets: kroA100, eli101 and kroA200.

A new hybrid gravitational particle swarm optimisation-ACO with local search mechanism, PSOGSA-ACO-Ls for TSP

The travelling salesman problem (TSP) is a hard combinatorial optimisation problem and a popular benchmarking problem at the same time. The TSP has also a number of practical real-world and industrial applications, such as routing in internet of things, IoT, networks, path planning in robotics and many others. In this paper, a new hybrid algorithm for the TSP is proposed; it combines gravitational particle swarm optimisation (PSOGSA) and ACO, and is called ant supervised by gravitational particle swarm optimisation with a local search, PSOGSA-ACO-LS. PSOGSA is used to optimise ACO settings while a local search mechanism, 2-Opt is employed by ACO to ameliorate its local solutions. The proposed method is evaluated using a set a test benches from the TSPLib database including: eil51, berlin52, st70, eil76, rat99, eil101, kroA100, and kroA200. Experimental results show that ACO-GPSO-LS is able to solve the set of TSP instances listed below including the large TSP data sets: kroA100, eli101 and kroA200.

Directed Dominating Set Problem Studied by Cavity Method: Warning Propagation and Population Dynamics**Supported by the Doctoral Startup Fund of Xinjiang University of China under Grant No. 208-61357 and the National Natural Science Foundation of China under Grant No. 11765021

The minimal dominating set for a digraph (directed graph) is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erdős-Rényi graphs to find the ground state energy. In this paper we continue to develope the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following. (i) The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one. (ii) We determine the ground state energy and the transition point of the Erdős-Rényi random graph. (iii) The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm.

Breakout variable neighbourhood search for the minimum interference frequency assignment problem

PurposeThis paper aims to describe two enhancements of the variable neighbourhood search (VNS) algorithm to solve efficiently the minimum interference frequency assignment problem (MI-FAP) which is a major issue in the radio networks, as well as a well-known NP-hard combinatorial optimisation problem. The challenge is to assign a frequency to each transceiver of the network with limited or no interferences at all. Indeed, considering that the number of radio networks users is ever increasing and that the radio spectrum is a scarce and expensive resource, the latter should be carefully managed to avoid any interference.Design/methodology/approachThe authors suggest two new enhanced VNS variants for MI-FAP, namely, the iterated VNS (It-VNS) and the breakout VNS (BVNS). These two algorithms were designed based on the hybridising and the collaboration approaches that have emerged as two powerful means to solve hard combinatorial optimisation problems. Therefore, these two methods draw their strength from other meta-heuristics. In addition, the authors introduced a new mechanism of perturbation to enhance the performance of VNS. An extensive experiment was conducted to evaluate the performance of the proposed methods on some well-known MI-FAP datasets. Moreover, they carried out a comparative study with other metaheuristics and achieved the Friedman’s non-parametric statistical test to check the actual effect of the proposed enhancements.FindingsThe experiments showed that the two enhanced methods (It-VNS) and (BVNS) achieved better results than the VNS method. The comparative study with other meta-heuristics showed that the results are competitive and very encouraging. The Friedman’s non-parametric statistical test reveals clearly that the results of the three methods (It-VNS, BVNS and VNS) are significantly different. The authors therefore carried out the Nemenyi’s post hoc test which allowed us to identify those differences. The impact of the operated change on both the It-VNS and BVNS was thus confirmed. The proposed BVNS is competitive and able to produce good results as compared with both It-VNS and VNS for MI-FAP.Research limitations/implicationsApproached methods and particularly newly designed ones may have some drawbacks that weaken the results, in particular when dealing with extensive data. These limitations should therefore be eliminated through an appropriate approach with a view to design appropriate methods in the case of large-scale data.Practical implicationsThe authors designed and implemented two new variants of the VNS algorithm before carrying out an exhaustive experimental study. The findings highlighted the potential opportunities of these two enhanced methods which could be adapted and applied to other combinatorial optimisation problems, real world applications or academic problems.Originality/valueThis paper aims at enhancing the VNS algorithm through two new approaches, namely, the It-VNS and the BVNS. These two methods were applied to the MI-FAP which is a crucial problem arising in a radio network. The numerical results are interesting and demonstrate the benefits of the proposed approaches in particular BVNS for MI-FAP.

Approximate Q-Learning for Stacking Problems with Continuous Production and Retrieval

ABSTRACTThis paper presents for the first time a reinforcement learning algorithm with function approximation for stacking problems with continuous production and retrieval. The stacking problem is a hard combinatorial optimization problem. It deals with the arrangement of items in a localized area, where they are organized into stacks to allow a delivery in a required order. Due to the characteristics of stacking problems, for example, the high number of states, reinforcement learning is an appropriate method since it allows learning in an unknown environment. We apply a Sarsa algorithm to real-world problem instances arising in steel industry. We use linear function approximation and elaborate promising characteristics of instances for this method. Further, we propose features that do not require specific knowledge about the environment and hence are applicable to any stacking problem with similar characteristics. In our experiments we show fast learning of the applied method and it’s suitability for real-world instances.

Ant colony system with a novel Non-DaemonActions procedure for multiprocessor task scheduling in multistage hybrid flow shop

Abstract This paper presents an ant colony system with a novel Non-DaemonActions procedure (ACSNDP) algorithm for multiprocessor task scheduling in multistage hybrid flow shop. A DaemonActions procedure is an optional component of ant colony optimization (ACO), which integrates problem-specific actions that cannot be performed by single ants such as the actions performed by local search routines. In many applications to hard combinatorial optimization problems, ACO performs best when integrated with local search routines because they move the ants' solutions to their local optimums. However, such an integration is shown as an effective approach only experimentally, and does not have any effects on the convergence properties of ACO theoretically, since the validity of convergence proofs depends only on the way solutions are constructed and not on the fact that the solutions are moved or not to their local optimums. Furthermore, it can be noticed that the traditional DaemonActions procedure does not interfere in the way solutions are constructed because local search routines have been always integrated with ACO in a daemon fashion i.e. they have been made hidden to the ants, and the ants do not how their solutions have been relocated. Consequently, the ants may perform limited exploitation (intensification) because they cannot exploit the problem-specific knowledge brought by the local search routine in such a way that enables them to construct these local optimums by themselves in the upcoming tour. In order to overcome these limitations, a novel Non-DaemonActions procedure, which can interfere positively in the way solutions are constructed, is proposed as follows. Iteratively at the end of each tour, the local search routine tries to improve the constructed solution, and then if a local optimum has been found, the ant learns the modifications made by the local search routine on its solution, and performs the corresponding modifications in the pheromone concentrations and heuristic information that will probably enable it to construct this local optimum by itself, in the upcoming tour, before the application of the local search over again. If the ant can do that, it will be able to construct more accurate local optimums in the upcoming tours by repeating the whole process over and over again, and thus enhance its exploitation capabilities. The proposed algorithm is tested on 700 well-known benchmark instances, with the proposed Non-DaemonActions procedure, and without it using the classical alternatives, and also compared with other 12 algorithms well-known in the literature. Computational results verify the improvements achieved by the proposed procedure, and show the superiority of the proposed algorithm over 7 of the compared works in terms of solution quality.

Data mining based framework to assess solution quality for the rectangular 2D strip-packing problem

In this paper, we explore the use of reference values (predictors) for the optimal objective function value of hard combinatorial optimization problems, instead of bounds, obtained by data mining techniques, and that may be used to assess the quality of heuristic solutions for the problem. With this purpose, we resort to the rectangular two-dimensional strip-packing problem (2D-SPP), which can be found in many industrial contexts. Mostly this problem is solved by heuristic methods, which provide good solutions. However, heuristic approaches do not guarantee optimality, and lower bounds are generally used to give information on the solution quality, in particular, the area lower bound. But this bound has a severe accuracy problem. Therefore, we propose a data mining-based framework capable of assessing the quality of heuristic solutions for the 2D-SPP. A regression model was fitted by comparing the strip height solutions obtained with the bottom-left-fill heuristic and 19 predictors provided by problem characteristics. Random forest was selected as the data mining technique with the best level of generalisation for the problem, and 30,000 problem instances were generated to represent different 2D-SPP variations found in real-world applications. Height predictions for new problem instances can be found in the regression model fitted. In the computational experimentation, we demonstrate that the data mining-based framework proposed is consistent, opening the doors for its application to finding predictions for other combinatorial optimisation problems, in particular, other cutting and packing problems. However, how to use a reference value instead of a bound, has still a large room for discussion and innovative ideas. Some directions for the use of reference values as a stopping criterion in search algorithms are also provided.

Hybrid estimation of distribution algorithms for solving a keyboard layout problem

ABSTRACTThe use of smartphones and handheld devices in our daily activities has sharply increased. The added features of wireless technology and related applications on these devices make it possible to write emails, notes, and long text. Even though the most commonly used electronic input device is a keyboard, very little work has been dedicated for finding an optimal layout for this device. In this research, the aim is to propose a better layout for the single-finger keyboard in terms of rapid typing. The keyboard layout problem can be formulated as a quadratic assignment problem, which is one of the hardest combinatorial optimization problems. Some well-known literary works in English are chosen for estimating the keying-in-time. A variant of genetic algorithm, namely, the estimation of distribution algorithms is used to find a better layout. The new layout is found to be efficient compared with some of the existing prominent keyboard layouts.

Omicron ACO. A New Ant Colony Optimization Algorithm


 
 
 Ant Colony Optimization (ACO) is a metaheuristic inspired by the foraging behavior of ant colonies that has been successful in the resolution of hard combinatorial optimization problems like the Traveling Salesman Problem (TSP). This paper proposes the Omicron ACO (OA), a novel population-based ACO alternative originally designed as an analytical tool. To experimentally prove OA advantages, this work compares the behavior between the OA and the MMAS as a function of time in two well-known TSP problems. A simple study of the behavior of OA as a function of its parameters shows its robustness.
 
 

Characterizing IRDP-instances of the skiving stock problem by means of polyhedral theory

ABSTRACTWe consider the one-dimensional skiving stock problem which is strongly related to the dual bin packing problem: given a supply of (small) items, the aim is to construct as many large items (specified by some target length) as possible. For this -hard combinatorial optimization problem, practical experience and computational simulations have shown that the continuous relaxation of the pattern-based ILP model provides very tight bounds. Although many instances are known to possess the integer round-down property (IRDP), there is only a very few number of corresponding theoretical results. In this article, we present a new approach to characterize IRDP-instances by means of polyhedral theory which, in a first step, provides an alternative proof for a well-known result of E.J. Zak. As a main contribution, we formulate the IRDP for two new classes of instances appearing in the context of the divisible case.

A Very Fast Heuristic for Combinatorial Optimization With Specific Application to Priority Rule Sequencing in Operations Management

This article presents mathematics of a generic polynomial-time heuristic which can be integrated into approaches for hard combinatorial optimization problems. The proposed method evaluates objects in a way that combines fuzzy reasoning with a greedy mechanism, thereby exploiting a fuzzy solution space using greedy methods. The effectiveness and efficiency of the proposed method are demonstrated on job-shop scheduling as one of the most challenging classical sequencing problems in the area of combinatorial optimization.

A Branch & Price algorithm for the minimum cost clique cover problem in max-point tolerance graphs

A point-interval $$(I_v, p_v)$$ is a pair constituted by an interval $$I_v$$ of $${\mathbb {R}}$$ and a point $$p_v \in I_v$$ . A graph $$G=(V,E)$$ is a Max-Point-Tolerance (MPT) graph if each vertex $$v\in V$$ can be mapped to a point-interval in such a way that (u, v) is an edge of G iff $$I_u \cap I_v \supseteq \{p_u, p_v\}$$ . MPT graphs constitute a superclass of interval graphs and naturally arise in genetic analysis as a way to represent specific relationships among DNA fragments extracted from a population of individuals. One of the most important applications of MPT graphs concerns the search for an association between major human diseases and chromosome regions from patients that exhibit loss of heterozygosity events. This task can be formulated as a minimum cost clique cover problem in a MPT graph and gives rise to a $${{\mathcal {N}}}{{\mathcal {P}}}$$ -hard combinatorial optimization problem known in the literature as the Parsimonious Loss of Heterozygosity Problem (PLOHP). In this article, we investigate ways to speed up the best known exact solution algorithm for the PLOHP as well as techniques to enlarge the size of the instances that can be optimally solved. In particular, we present a Branch&Price algorithm for the PLOHP and we develop a number of preprocessing techniques and decomposition strategies to dramatically reduce the size of its instances. Computational experiments show that the proposed approach is 10–30 $$\times $$ faster than previous approaches described in the literature, and suggest new directions for the development of future exact solution approaches that may prove of fundamental assistance in practice.

Scheduling the Italian National Volleyball Tournament

In this paper, we present our approach to sports scheduling, a process that led the Italian Volleyball League to adopt our calendar for the 2016–2017 and subsequent seasons. Sports scheduling is a hard combinatorial optimization problem whose solution requires modeling many different aspects of sports, some of which are unique to each sport or nation. The capability of producing a high-quality schedule is important for both balancing undesirable matches among teams and ensuring adequate coverage by television and other media. Through strong interaction with the Italian Volleyball League, we modeled and solved the problem to optimality using standard mathematical programming solvers. We also tested our solution using previous seasons’ tournaments to prove the capability of our model.

Priority-based and conflict-avoidance heuristics for multi-satellite scheduling

In this paper we address the problem of multi-satellite scheduling with limited observing ability. As with other computationally hard combinatorial optimization problems, a two-stage heuristic method is developed to obtain high quality solutions in a reasonable amount of computation time. The first stage involves the determination of an observing sequence and the generation of a feasible scheduling scheme. We propose several priority-based and conflict-avoidance heuristic strategies and develop the time-based greedy approaches, the weight-based greedy approaches, and an improved differential evolution (DE) algorithm. The second stage consists of further improvement strategies under different resource contentions, thus improving the scheduling results further. Finally, we design different classes of instances to test the efficiency and applicability of the methods. Computational results reveal that the new proposed methods routinely delivered very close to optimal solutions.

A biased random-key genetic algorithm for the project scheduling problem with flexible resources

In this paper, we investigate a resource-constrained project scheduling problem with flexible resources. This is an $$\mathcal {NP}$$ -hard combinatorial optimization problem that consists of scheduling a set of activities requiring specific resource units of several skills. The goal is to minimize the makespan of the project. We propose a biased random-key genetic algorithm for computing feasible solutions for the referred problem. We study different decoding mechanisms: an already existing method in the literature, a new adapted serial scheduling generation scheme, and a combination of both. The new procedure is tested using a set of benchmark instances of the problem. The results provide strong evidence that the new heuristic is robust and yields high-quality feasible solutions.