Difficult Combinatorial Problems Research Articles

Using symmetry and evolutionary search to minimize sorting networks

Sorting networks are an interesting class of parallel sorting algorithms with applications in multiprocessor computers and switching networks. They are built by cascading a series of comparison-exchange units called comparators. Minimizing the number of comparators for a given number of inputs is a challenging optimization problem. This paper presents a two-pronged approach called Symmetry and Evolution based Network Sort Optimization (SENSO) that makes it possible to scale the solutions to networks with a larger number of inputs than previously possible. First, it uses the symmetry of the problem to decompose the minimization goal into subgoals that are easier to solve. Second, it minimizes the resulting greedy solutions further by using an evolutionary algorithm to learn the statistical distribution of comparators in minimal networks. The final solutions improve upon half-century of results published in patents, books, and peer-reviewed literature, demonstrating the potential of the SENSO approach for solving difficult combinatorial problems.

Solving the weighted MAX-SAT problem using the dynamic convexized method

Satisfiability (SAT) and maximum satisfiability (MAX-SAT) are difficult combinatorial problems that have many important real-world applications. In this paper we investigate the performance of the dynamic convexized method based heuristics on the weighted MAX-SAT problem. We first present an auxiliary function which is constructed based on a penalty function, and minimize the function by a local search method which can escape successfully from previously converged local minimizers by increasing the value of a parameter. Two algorithms of the approach are implemented and compared with the Greedy Randomized Adaptive Search Procedure (GRASP) and the GRASP with Path Relinking (GRASP + PR). Experimental results illustrate efficient and faster convergence of our two algorithms.

Cost optimisation of supply chain networks using Ant Colony Optimisation

With increasing competitiveness in the business world, the focus on supply chains is receiving more attention. Therefore, the supply chain has to be made more effective by reducing unnecessary losses. These losses are caused due to production, distribution planning and improper routing of vehicles in supply chain networks. The objective of this paper is to reduce costs across the supply chain by effectively allocating distribution centres to warehouses, reducing transportation costs and inventory costs. A non-traditional optimisation tool that can effectively find good solutions to difficult combinatorial problems is Ant Colony Optimisation (ACO). ACO is a meta-heuristic that generates information about the optimisation procedure in the form of a pheromone matrix. This information can be shared and used by members of the colony. This provides a platform to manage the supply chain optimally. The ants start from the warehouse and travel to various distribution centres which are assigned to the respective warehouses for distribution. A pheromone matrix was developed based on the input information from all the ants. Constraints were imposed on the routes traversed by ants. The constraints given for ants are warehouse capacity and maximum distance to be travelled by ants.

Bee colony optimization for the p-center problem

Bee colony optimization (BCO) is a relatively new meta-heuristic designed to deal with hard combinatorial optimization problems. It is biologically inspired method that explores collective intelligence applied by the honey bees during nectar collecting process. In this paper we apply BCO to the p-center problem in the case of symmetric distance matrix. On the contrary to the constructive variant of the BCO algorithm used in recent literature, we propose variant of BCO based on the improvement concept (BCOi). The BCOi has not been significantly used in the relevant BCO literature so far. In this paper it is proved that BCOi can be a very useful concept for solving difficult combinatorial problems. The numerical experiments performed on well-known benchmark problems show that the BCOi is competitive with other methods and it can generate high-quality solutions within negligible CPU times.

Structural Similarity: Spectral Methods for Relaxed Blockmodeling

In this paper we propose the concept of structural similarity as a relaxation of blockmodeling in social network analysis. Most previous approaches attempt to relax the constraints on partitions, for instance, that of being a structural or regular equivalence to being approximately structural or regular, respectively. In contrast, our approach is to relax the partitions themselves: structural similarities yield similarity values instead of equivalence or non-equivalence of actors, while strictly obeying the requirement made for exact regular equivalences. Structural similarities are based on a vector space interpretation and yield efficient spectral methods that, in a more restrictive manner, have been successfully applied to difficult combinatorial problems such as graph coloring. While traditional blockmodeling approaches have to rely on local search heuristics, our framework yields algorithms that are provably optimal for specific data-generation models. Furthermore, the stability of structural similarities can be well characterized making them suitable for the analysis of noisy or dynamically changing network data.

A rough set approach to feature selection based on ant colony optimization

Rough set theory is one of the effective methods to feature selection, which can preserve the meaning of the features. The essence of rough set approach to feature selection is to find a subset of the original features. Since finding a minimal subset of the features is a NP-hard problem, it is necessary to investigate effective and efficient heuristic algorithms. Ant colony optimization (ACO) has been successfully applied to many difficult combinatorial problems like quadratic assignment, traveling salesman, scheduling, etc. It is particularly attractive for feature selection since there is no heuristic information that can guide search to the optimal minimal subset every time. However, ants can discover the best feature combinations as they traverse the graph. In this paper, we propose a new rough set approach to feature selection based on ACO, which adopts mutual information based feature significance as heuristic information. A novel feature selection algorithm is also given. Jensen and Shen proposed a ACO-based feature selection approach which starts from a random feature. Our approach starts from the feature core, which changes the complete graph to a smaller one. To verify the efficiency of our algorithm, experiments are carried out on some standard UCI datasets. The results demonstrate that our algorithm can provide efficient solution to find a minimal subset of the features.

ADAPTIVENESS IN MONOTONE PSEUDO-BOOLEAN OPTIMIZATION AND STOCHASTIC NEURAL COMPUTATION

Hopfield neural network (HNN) is a nonlinear computational model successfully applied in finding near-optimal solutions of several difficult combinatorial problems. In many cases, the network energy function is obtained through a learning procedure so that its minima are states falling into a proper subspace (feasible region) of the search space. However, because of the network nonlinearity, a number of undesirable local energy minima emerge from the learning procedure, significantly effecting the network performance. In the neural model analyzed here, we combine both a penalty and a stochastic process in order to enhance the performance of a binary HNN. The penalty strategy allows us to gradually lead the search towards states representing feasible solutions, so avoiding oscillatory behaviors or asymptotically instable convergence. Presence of stochastic dynamics potentially prevents the network to fall into shallow local minima of the energy function, i.e., quite far from global optimum. Hence, for a given fixed network topology, the desired final distribution on the states can be reached by carefully modulating such process. The model uses pseudo-Boolean functions both to express problem constraints and cost function; a combination of these two functions is then interpreted as energy of the neural network. A wide variety of NP-hard problems fall in the class of problems that can be solved by the model at hand, particularly those having a monotonic quadratic pseudo-Boolean function as constraint function. That is, functions easily derived by closed algebraic expressions representing the constraint structure and easy (polynomial time) to maximize. We show the asymptotic convergence properties of this model characterizing its state space distribution at thermal equilibrium in terms of Markov chain and give evidence of its ability to find high quality solutions on benchmarks and randomly generated instances of two specific problems taken from the computational graph theory.

Evolutionary Approaches to the Linear Machine Layout Problem

Flexible Manufacturing Systems (FMSs) cope with multi-product, usually small sized production. In this research work we investigate the use of evolutionary methods to solve the linear or single-row layout problem, which is the most commonly implemented layout in FMSs. Three different approaches, based on Ant Colony Optimization, Genetic Algorithms and Particle Swarm Optimization are tested. The experimental results show that a near optimal solution can be found for all three methods, exploiting only a small portion of the feasible solution space, pinpointing once more the merit of using evolutionary algorithms to tackle difficult combinatorial problems.

Construction imitating ants: Resource-unconstrained scheduling with artificial ants

The paper presents a methodology to arrive at critical path calculations in construction networks by imitating the natural selection processes utilized by real-life ants in search of shortest paths to a food source, and by using Ant Colony Optimization (ACO) algorithms. Ant Colony Optimization is a population-based, artificial multi-agent, general-search technique for the solution of difficult combinatorial problems with its theoretical roots based on the behavior of real ant colonies. The fundamental mathematical background of the ACO method is outlined and a suggested possible implementation strategy is described for solving for longest (critical) paths in construction schedule networks. The ACO methodology should be of interest to both researchers and practitioners since it provides an alternative method to critical path calculations, with a wide range of applications. The described ACO virtual multi-agent approach is supplemented by a sample case study as well as algorithms for the solution of resource-unconstrained construction schedules.

Interleaving between Ant Colony Optimization and Tabu Search for Image Matching

Image matching plays an important role in many applications such as multi-modality medical imaging and multi-spectral image analysis. The role of matching is to integrate multiple sources of object information into a single image. The matching problem consists of determining the unknown transform parameters required to map one image to match the other image(20). Different non – traditional methods are used for solving this kind of problem. Among these methods are the Genetic Algorithms, Neural Networks & Simulating Annealing. Swarm Intelligence (SI) algorithms take their inspiration from the collective behavior of natural, for example, ant colonies, flocks of birds, or fish shoals, a particularly successful strandant colony optimization (ACO)(1). Ant Colony Optimization is a population-based general search technique, proposed by Dorigo(1992,1996), for the solution of difficult combinatorial problems)4). The studies show that, in nature, the ant colony is able to discover the shortest paths between the nest and food sources very efficiently, such a deposit substance is called pheromone during talking and another ants can smell it, if one of ants find a short path, it feedback on the same path and the value of pheromone on this path increases and a another ants gradually chose this path.(22) Tabu search is one of the best known heuristic to choose the next neighbor to move on. At each step, one chooses the best neighbor with respect to specific function (23). The basic idea in this paper is using Ant Colony Optimization(ACO) & Tabu Search(TS) as a success strategy for matching two images. The suggestion algorithm evaluation is a good promising solution, by providing an optimal algorithm which is executed by optimal time and coast, I believe that there is no prior research conjoining the two topics in this way. The program is written in Matlab language (6.5).

Biomolecular computing: Is it ready to take off?

Biomolecular computing is an emerging field at the interface of computer science, biological science and engineering. It uses DNA and other biological materials as the building blocks for construction of living computational machines to solve difficult combinatorial problems. In this article, notable advances in the biomolecular computing are reviewed and challenges associated with this multidisciplinary research are addressed. Finally, several perspectives are given based on the review of biomolecular computing.

Randomised restarted search in ILP

Recent statistical performance studies of search algorithms in difficult combinatorial problems have demonstrated the benefits of randomising and restarting the search procedure. Specifically, it has been found that if the search cost distribution of the non-restarted randomised search exhibits a slower-than-exponential decay (that is, a “heavy tail”), restarts can reduce the search cost expectation. We report on an empirical study of randomised restarted search in ILP. Our experiments conducted on a high-performance distributed computing platform provide an extensive statistical performance sample of five search algorithms operating on two principally different classes of ILP problems, one represented by an artificially generated graph problem and the other by three traditional classification benchmarks (mutagenicity, carcinogenicity, finite element mesh design). The sample allows us to (1) estimate the conditional expected value of the search cost (measured by the total number of clauses explored) given the minimum clause score required and a “cutoff” value (the number of clauses examined before the search is restarted), (2) estimate the conditional expected clause score given the cutoff value and the invested search cost, and (3) compare the performance of randomised restarted search strategies to a deterministic non-restarted search. Our findings indicate striking similarities across the five search algorithms and the four domains, in terms of the basic trends of both the statistics (1) and (2). Also, we observe that the cutoff value is critical for the performance of the search algorithm, and using its optimal value in a randomised restarted search may decrease the mean search cost (by several orders of magnitude) or increase the mean achieved score significantly with respect to that obtained with a deterministic non-restarted search.

MUTUAL INFORMATION CLUSTERING FOR EFFICIENT MINING OF FUZZY ASSOCIATION RULES WITH APPLICATION TO GENE EXPRESSION DATA ANALYSIS

Fuzzy association rules can reveal useful dependencies and interactions hidden in large gene expression data sets. However their derivation perplexes very difficult combinatorial problems that depend heavily on the size of these sets. The paper follows a divide and conquer approach to the problem that obtains computationally manageable solutions. Initially we cluster genes that more probably are associated. Thereafter, the fuzzy association rule extraction algorithms confront many but significantly reduced computationally problems that usually can be processed very fast. The clustering phase is accomplished by means of an approach based on mutual information (MI). This approach uses the mutual information as a similarity measure. However, the numerical evaluation of the MI is subtle. We experiment with the main methods and we compare between them. As the device that implements the mutual information clustering we use a SOM (Self-Organized Map) based approach that is capable of effectively incorporating supervised bias. After the mutual information clustering phase the fuzzy association rules are extracted locally on a per cluster basis. The paper presents an application of the techniques for mining the gene expression data. However, the presented techniques can easily be adapted and can be fruitful for intelligent exploration of any other similar data set as well.

An LP-based heuristic procedure for the generalized assignment problem with special ordered sets

The generalized assignment problem with special ordered sets (GAPS2), is the problem of allocating n tasks to m time-periods, where each task must be assigned to a time-period, or shared between two consecutive time-periods. For reasonably large values of m and n the NP-hard combinatorial problem GAPS2 becomes intractable for standard mathematical programming software, hence there is a need for heuristic algorithms to solve such problems. It will be shown how an LP-based heuristic developed previously for the well-established generalized assignment problem can be modified and extended to solve GAPS2. Encouraging results, in terms of speed and accuracy, in particular when compared to an existing heuristic for GAPS2, are described. Scope and purpose This paper considers an extension of the hard combinatorial problem—the generalized assignment problem (GAP). GAP is the problem of assigning to best advantage a set of jobs to a set of machines where each job must be assigned to exactly one machine, but each machine, which has limited capacity, may have more than one job assigned to it. For practical cases the terms ‘jobs’ and ‘machines’ are just indicative of the sets of objects to be assigned and the sets of objects to which they are to be assigned. In the extension to GAP considered in this paper, the assignment is of tasks to time periods, and tasks are permitted to be split across two periods, if required, provided the time periods are consecutive. This problem is termed GAPS2. A heuristic is developed to solve GAPS2 after it is established that there is unlikely to be any exact method which will solve the problem. The heuristic turns out to be very effective and the results of computational testing indicate that near-optimal solutions can be derived rapidly to instances of GAPS2 of substantial size. Although the linear programming based heuristic is relatively straightforward, the contribution of this paper is to show that very good quality solutions can be obtained rapidly to difficult combinatorial problems of substantial size. The described heuristic considerably outperforms the only other published heuristic for GAPS2.

Scheduling of an assembly line with a multi-objective genetic algorithm

Scheduling problems are difficult combinatorial problems because of the extremely large search space of possible solutions and the large number of local optima that arise. A multi-objective genetic algorithm is presented as an intelligent algorithm for scheduling of the mixed-model assembly line in this paper. The Pareto ranking method and distance-dispersed approach are employed to evaluate the fitness of the individuals. The computational results show that the proposed multi-objective genetic algorithm is quite effective.

Configuration landscape analysis and backbone guided local search.: Part I: Satisfiability and maximum satisfiability

Boolean satisfiability (SAT) and maximum satisfiability (Max-SAT) are difficult combinatorial problems that have many important real-world applications. In this paper, we first investigate the configuration landscapes of local minima reached by the WalkSAT local search algorithm, one of the most effective algorithms for SAT. A configuration landscape of a set of local minima is their distribution in terms of quality and structural differences relative to an optimal or a reference solution. Our experimental results show that local minima from WalkSAT form large clusters, and their configuration landscapes constitute big valleys, in that high quality local minima typically share large partial structures with optimal solutions. Inspired by this insight into WalkSAT and the previous research on phase transitions and backbones of combinatorial problems, we propose and develop a novel method that exploits the configuration landscapes of such local minima. The new method, termed as backbone-guided search, can be embedded in a local search algorithm, such as WalkSAT, to improve its performance. Our experimental results show that backbone-guided local search is effective on overconstrained random Max-SAT instances. Moreover, on large problem instances from a SAT library (SATLIB), the backbone guided WalkSAT algorithm finds satisfiable solutions more often than WalkSAT on SAT problem instances, and obtains better solutions than WalkSAT on Max-SAT problem instances, improving solution quality by 20% on average.

Defining tabu list size and aspiration criterion within tabu search methods

An investigation to explicitly define two key elements in tabu search methods is attempted. In this study a functional representation of the tabu list size is presented and a softer aspiration criterion is put forward. Experiments are conducted on a set of p-median problems. Scope and purpose Tabu search is a metaheuristic that proved successful in finding good solutions to difficult combinatorial problems that were hard to find otherwise. In this study, we attempt to help the user in the choice of some of the parameters used in this type of heuristics. We based our analysis on the tabu list size and on an implementation on how to define the aspiration criterion. This added information can be valuable to those users who apply these methods in a near systematic manner without relying heavily on experimentations. As an example we used a simple location problem to test the usefulness of these ideas.

Evolutionary algorithms for flowshop sequencing with non‐unique jobs

AbstractSequencing problems are difficult combinatorial problems because of the extremely large search space of possible solutions and the large number of “local” optima that arise. Unlike other NP‐hard combinatorial problems, the search space, in general, for sequencing problems (under the makespan objective) consists of sequences with objective function values that lie within only a relatively small amount of each other. This means that when a change is made to the sequence, an improvement or non‐improvement is not easily recognised. This makes the problem much more difficult to solve. A number of constructive heuristics exist that obtain good solutions in a short period of time, however, the output of such algorithms is generally a single sequence which may not be feasible or preferred with respect to industry constraints. Other heuristic algorithms such as Simulated Annealing (SA) and Tabu Search (TS) have also been applied and successes have been reported. However, the performance is dependent upon a number of finely tuned parameters and the output is again only a single solution. For these reasons, Evolutionary Algorithms (EAs) may be suitable solution strategy, for which limited research has been performed. In this research, a number of new EAs have been proposed and a number of modifications have been made to several constructive algorithms to cope with non‐unique jobs or jobs with multiple demands. A numerical comparison of a number of benchmark problems and real data of a truck assembly line has also been presented.

Evolutionary algorithms for flowshop sequencing with non-unique jobs

Sequencing problems are difficult combinatorial problems because of the extremely large search space of possible solutions and the large number of “local” optima that arise. Unlike other NP-hard combinatorial problems, the search space, in general, for sequencing problems (under the makespan objective) consists of sequences with objective function values that lie within only a relatively small amount of each other. This means that when a change is made to the sequence, an improvement or non-improvement is not easily recognised. This makes the problem much more difficult to solve. A number of constructive heuristics exist that obtain good solutions in a short period of time, however, the output of such algorithms is generally a single sequence which may not be feasible or preferred with respect to industry constraints. Other heuristic algorithms such as Simulated Annealing (SA) and Tabu Search (TS) have also been applied and successes have been reported. However, the performance is dependent upon a number of finely tuned parameters and the output is again only a single solution. For these reasons, Evolutionary Algorithms (EAs) may be suitable solution strategy, for which limited research has been performed. In this research, a number of new EAs have been proposed and a number of modifications have been made to several constructive algorithms to cope with non-unique jobs or jobs with multiple demands. A numerical comparison of a number of benchmark problems and real data of a truck assembly line has also been presented.

Voltage drop and power losses fast evaluation through equivalent models of feeders for optimal operation of automated distribution networks

AbstractIn the paper, a generalised distribution‐feeder equivalent model is developed and presented. The model has been specifically developed aiming at applications in the field of optimal operation of automated distribution networks. It allows to easily and rapidly perform calculations for optimal reconfiguration and/or compensation. These kind of difficult combinatorial problems have indeed been extensively solved in literature by means of heuristic strategies and non‐deterministic algorithms. These methodologies are intrinsically iterative, requiring many evaluations of the objective function, which is often computationally time‐consuming. The studied elementary system is a distribution feeder with lumped loads and capacitor banks. The proposed model is bi‐directional, because this condition is required when simulating reconfiguration. The paper first shows the derivation of the equivalent models parameters when considering a single branch element, then, a calculation methodology for radial networks is presented. In this case, the reduction strategy requires a preliminary visit of all the nodes from the terminal ones up to the root of the tree‐network and in turn considering the summed up loads below, as terminal loads. The proposed networks reduction can be applied for fast implementation of power losses and voltage drops calculations when using advanced dynamic data structure for the networks representation. The proposed methodology has then been implemented by means of these information tools and tested on different distribution systems. so as to evaluate the errors due to the feeders and networks reduction.