Algorithms For Combinatorial Optimization Problems Research Articles

Contributors

Contributors

A hybrid approach using chaotic dynamics and global search algorithms for combinatorial optimization problems

Chaotic dynamics have been effectively applied to improve various heuristic algorithms for combinatorial optimization problems in many studies. Currently, the most used chaotic optimization scheme is to drive heuristic solution search algorithms applicable to large-scale problems by chaotic neurodynamics including the tabu effect of the tabu search. Alternatively, meta-heuristic algorithms are used for combinatorial optimization by combining a neighboring solution search algorithm, such as tabu, gradient, or other search method, with a global search algorithm, such as genetic algorithms (GA), ant colony optimization (ACO), or others. In these hybrid approaches, the ACO has effectively optimized the solution of many benchmark problems in the quadratic assignment problem library. In this paper, we propose a novel hybrid method that combines the effective chaotic search algorithm that has better performance than the tabu search and global search algorithms such as ACO and GA. Our results show that the proposed chaotic hybrid algorithm has better performance than the conventional chaotic search and conventional hybrid algorithms. In addition, we show that chaotic search algorithm combined with ACO has better performance than when combined with GA.

Statistical Design of Genetic Algorithms for Combinatorial Optimization Problems

Many genetic algorithms (GA) have been applied to solve different NP‐complete combinatorial optimization problems so far. The striking point of using GA refers to selecting a combination of appropriate patterns in crossover, mutation, and and so forth and fine tuning of some parameters such as crossover probability, mutation probability, and and so forth. One way to design a robust GA is to select an optimal pattern and then to search for its parameter values using a tuning procedure. This paper addresses a methodology to both optimal pattern selection and the tuning phases by taking advantage of design of experiments and response surface methodology. To show the performances of the proposed procedure and demonstrate its applications, it is employed to design a robust GA to solve a project scheduling problem. Through the statistical comparison analyses between the performances of the proposed method and an existing GA, the effectiveness of the methodology is shown.

Focus on Authors

Focus on Authors

Adaptive Local Ratio

Local ratio is a well-known paradigm for designing approximation algorithms for combinatorial optimization problems. At a very high level, a local-ratio algorithm first decomposes the input weight function w into a positive linear combination of simpler weight functions or models. Guided by this process, a solution S is constructed such that S is $\alpha$-approximate with respect to each model used in the decomposition. As a result, S is $\alpha$-approximate under w as well. These models usually have a very simple structure that remains “unchanged” throughout the execution of the algorithm. In this work we show that adaptively choosing a model from a richer spectrum of functions can lead to a better local ratio. Indeed, by turning the search for a good model into an optimization problem of its own, we get improved approximations for a data migration problem.

A Multiagent Evolutionary Algorithm for Combinatorial Optimization Problems

Based on our previous works, multiagent systems and evolutionary algorithms (EAs) are integrated to form a new algorithm for combinatorial optimization problems (CmOPs), namely, MultiAgent EA for CmOPs (MAEA-CmOPs). In MAEA-CmOPs, all agents live in a latticelike environment, with each agent fixed on a lattice point. To increase energies, all agents compete with their neighbors, and they can also increase their own energies by making use of domain knowledge. Theoretical analyses show that MAEA-CmOPs converge to global optimum solutions. Since deceptive problems are the most difficult CmOPs for EAs, in the experiments, various deceptive problems with strong linkage, weak linkage, and overlapping linkage, and more difficult ones, namely, hierarchical problems with treelike structures, are used to validate the performance of MAEA-CmOPs. The results show that MAEA-CmOP outperforms the other algorithms and has a fast convergence rate. MAEA-CmOP is also used to solve large-scale deceptive and hierarchical problems with thousands of dimensions, and the experimental results show that MAEA-CmOP obtains a good performance and has a low computational cost, which the time complexity increases in a polynomial basis with the problem size.

Application of ant colony, genetic algorithm and data mining-based techniques for scheduling

In this paper, we have proposed a novel use of data mining algorithms for the extraction of knowledge from a large set of flow shop schedules. The purposes of this work is to apply data mining methodologies to explore the patterns in data generated by an ant colony algorithm performing a scheduling operation and to develop a rule set scheduler which approximates the ant colony algorithm's scheduler. Ant colony optimization (ACO) is a paradigm for designing metaheuristic algorithms for combinatorial optimization problems. The natural metaphor on which ant algorithms are based is that of ant colonies. Fascinated by the ability of the almost blind ants to establish the shortest route from their nests to the food source and back, researchers found out that these ants secrete a substance called ‘pheromone’ and use its trails as a medium for communicating information among each other. The ant algorithm is simple to implement and results of the case studies show its ability to provide speedy and accurate solutions. Further, we employed the genetic algorithm operators such as crossover and mutation to generate the new regions of solution. The data mining tool we have used is Decision Tree, which is produced by the See5 software after the instances are classified. The data mining is for mining the knowledge of job scheduling about the objective of minimization of makespan in a flow shop environment. Data mining systems typically uses conditional relationships represented by IF-THEN rules and allowing the production managers to easily take the decisions regarding the flow shop scheduling based on various objective functions and the constraints.

A simple greedy algorithm for a class of shuttle transportation problems

Greedy algorithms for combinatorial optimization problems are typically direct and efficient, but hard to prove optimality. The paper presents a special class of transportation problems where a supplier sends goods to a set of customers, returning to the source after each delivery. We show that these problems with different objective functions share a common structural property, and therefore a simple but powerful generic greedy algorithm yields optimal solutions for all of them.

A comparison of classical and quantum annealing dynamics

Simulated annealing and quantum annealing are algorithms for combinatorial optimization problems. The former brings solutions using thermal fluctuations and through classical dynamics, while the latter does using quantum fluctuations and through quantum dynamics. In this paper, dynamics of these two algorithms are compared by means of the Kibble-Zurek argument, employing a one-dimensional random Ising model as an exercise for algorithms. We reveal that quantum annealing reduces residual errors faster than simulated annealing with decreasing annealing rate. The result implies the advantage of quantum annealing over simulated annealing.

Models and algorithms for combinatorial optimization problems arising in railway applications

This is a summary of the author’s PhD thesis supervised by Alberto Caprara and Paolo Toth and defended on 29 May 2007 at the Universita di Bologna. The thesis is written in English and is available from the author upon request. This work deals with Railway Optimization, and in particular it focuses on the Train Timetabling Problem (in the basic version on a corridor and in the extension to a railway network), and on the Train Unit Assignment Problem. Integer Linear Programming (ILP) formulations are proposed for both problems, and their continuous and Lagrangian relaxations are used to obtain optimal and heuristic solutions to real-world instances.

-Similar Elimination to Enhance Search Performance of Multiobjective Evolutionary Algorithms

In this paper, we propose δ-similar elimination to improve the search performance of multiobjective evolutionary algorithms in combinatorial optimization problems. This method eliminates similar individuals in objective space to fairly distribute selection among the different regions of the instantaneous Pareto front. We investigate four eliminating methods analyzing their effects using NSGA-II. In addition, we compare the search performance of NSGA-II enhanced by our method and NSGA-II enhanced by controlled elitism.

Faster communication in known topology radio networks

This paper concerns the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in known topology radio networks, i.e., where for each primitive the schedule of transmissions is precomputed in advance based on full knowledge about the size and the topology of the network. The first part of the paper examines the two communication primitives in arbitrary graphs. In particular, for the broadcast task we deliver two new results: a deterministic efficient algorithm for computing a radio schedule of length D + O(log3 n), and a randomized algorithm for computing a radio schedule of length D + O(log2 n). These results improve on the best currently known D + O(log4 n) time schedule due to Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222–231, 2005). Later we propose a new (efficiently computable) deterministic schedule that uses 2D + Δlog n + O(log3 n) time units to complete the gossiping task in any radio network with size n, diameter D and max-degree Δ. Our new schedule improves and simplifies the currently best known gossiping schedule, requiring time $$O(D+\sqrt[{i+2}]{D}\Delta\log^{i+1} n)$$ , for any network with the diameter D = Ω(log i+4 n), where i is an arbitrary integer constant i ≥ 0, see Gąsieniec et al. (Proceedings of the 11th International Colloquium on Structural Information and Communication Complexity, vol. 3104, pp. 173–184, 2004). The second part of the paper focuses on radio communication in planar graphs, devising a new broadcasting schedule using fewer than 3D time slots. This result improves, for small values of D, on the currently best known D + O(log3 n) time schedule proposed by Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222–231, 2005). Our new algorithm should be also seen as a separation result between planar and general graphs with small diameter due to the polylogarithmic inapproximability result for general graphs by Elkin and Kortsarz (Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, vol. 3122, pp. 105–116, 2004; J. Algorithms 52(1), 8–25, 2004).

Objective Function Adjustment Algorithm for Combinatorial Optimization Problems

An improved algorithm of Guided Local Search called objective function adjustment algorithm is proposed for combinatorial optimization problems. The performance of Guided Local Search is improved by objective function adjustment algorithm using multipliers which can be adjusted during the search process. Moreover, the idea of Tabu Search is introduced into the objective function adjustment algorithm to further improve the performance. The simulation results based on some TSPLIB benchmark problems showed that the objective function adjustment algorithm could find better solutions than Local Search, Guided Local Search and Tabu Search.

The Bidimensional Theory of Bounded-Genus Graphs

Bidimensionality provides a tool for developing subexponential fixed-parameter algorithms for combinatorial optimization problems on graph families that exclude a minor. This paper extends the theory of bidimensionality for graphs of bounded genus (which is a minor-excluding family). Specifically we show that, for any problem whose solution value does not increase under contractions and whose solution value is large on a grid graph augmented by a bounded number of handles, the treewidth of any bounded-genus graph is at most a constant factor larger than the square root of the problem's solution value on that graph. Such bidimensional problems include vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, r-dominating set, connected dominating set, planar set cover, and diameter. On the algorithmic side, by showing that an augmented grid is the prototype bounded-genus graph, we generalize and simplify many existing algorithms for such problems in graph classes excluding a minor. On the combinatorial side, our result is a step toward a theory of graph contractions analogous to the seminal theory of graph minors by Robertson and Seymour.

Metaheuristic algorithms for combinatorial optimization problems

We present the main results in the author’s Ph.D. thesis (Iori 2004), defended at the University of Bologna in April 2004 and supervised by S. Martello. The thesis is written in English and is available from the author upon request. It proposes exact and metaheuristic algorithms for solving some relevant combinatorial optimization problems, with particular emphasis on scheduling, two-dimensional cutting and packing and capacitated vehicle routing. The performance of each algorithm is tested through extensive computational experiments and comparison with other approaches in the literature.

AN OVERVIEW OF SOME HEURISTIC ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS

Heuristic algorithms (or simply heuristics) are methods that seek for high quality solutions within a reasonable (limited) amount of time without being able to guarantee optimality. They often come out as a result of imitation of the real world (physics, nature, biology, etc.). In this paper, we give an overview of some heuristic algorithms for combinatorial optimization problems. At the beginning, some definitions related to combinatorial optimization, as well as the principle (framework) and basic features of the heuristics for combinatorial problems are concerned. Then, several popular heuristic algorithms are discussed, namely: descent local search, simulated annealing, tabu search, genetic algorithms, ant algorithms, and iterated local search. The unified paradigms of these heuristics are given. Finally, we present some results of comparisons of these algorithms for the well-known combinatorial problem, the quadratic assignment problem.

An algorithmic framework for improving heuristic solutions: Part I. A deterministic discount coupon traveling salesman problem

The efforts to develop solution algorithms for combinatorial optimization problems can be classified broadly into two categories, heuristic and rigorous approaches. These two approaches have ever-conflicting advantages and disadvantages in their computational load and solution quality. Due to the computational infeasibility of the rigorous approach for many practical combinatorial applications problem-specific heuristics are prevalent, even though they cannot provide any guarantee on the solution’s quality. In this paper, we propose a novel way to improve upon a solution obtained from heuristics by applying dynamic programming to the subset of the states visited during simulation of the heuristics. The method represents a way to take a family of solutions and patch them together as an improved solution. However, ‘patching’ is accomplished in state space rather than in solution space. We develop and apply the approach to a new traveling salesman problem (TSP) variant, which illustrates the important notions of optional and conditional tasks in planning and scheduling applications, to examine the degree of improvement that can be obtained by the method. For a small problem, we compare the quality of the solution with the globally optimal one. The proposed method can be generalized to other planning and scheduling problems as long as a reasonable set of heuristics exist for their solution.

Approximate Local Search in Combinatorial Optimization

Local search algorithms for combinatorial optimization problems are in general of pseudopolynomial running time and polynomial-time algorithms are often not known for finding locally optimal solutions for NP-hard optimization problems. We introduce the concept of epsilon-local optimality and show that an epsilon-local optimum can be identified in time polynomial in the problem size and 1/epsilon whenever the corresponding neighborhood can be searched in polynomial time, for epsilon > 0. If the neighborhood can be searched in polynomial time for a delta-local optimum, we present an algorithm that produces a (delta+epsilon)-local optimum in time polynomial in the problem size and 1/epsilon. As a consequence, a combinatorial optimization problem has a fully polynomial-time approximation scheme if and only if it has a fully polynomial-time augmentation schem

Approximate Local Search in Combinatorial Optimization

Local search algorithms for combinatorial optimization problems are in general of pseudopolynomial running time and polynomial-time algorithms are often not known for finding locally optimal solutions for NP-hard optimization problems. We introduce the concept of epsilon-local optimality and show that an epsilon-local optimum can be identified in time polynomial in the problem size and 1/epsilon whenever the corresponding neighborhood can be searched in polynomial time, for epsilon > 0. If the neighborhood can be searched in polynomial time for a delta-local optimum, we present an algorithm that produces a (delta+epsilon)-local optimum in time polynomial in the problem size and 1/epsilon. As a consequence, a combinatorial optimization problem has a fully polynomial-time approximation scheme if and only if it has a fully polynomial-time augmentation scheme.

New Rollout Algorithms for Combinatorial Optimization Problems

Rollout algorithms are new computational approaches used to determine near-optimal solutions for deterministic and stochastic combinatorial optimization problems. They are built on a generic base heuristic with the aim to construct another hopefully improved heuristic. However, rollout algorithms can be very expensive from the computational point of view, so their use for practical applications can be limited. In this article, we propose modified versions of the rollout algorithms to solve deterministic optimization problems, defined in such a way to limit the computational cost, without worsening the quality of the final approximate solution obtained.