Representative Combinatorial Optimization Problems Research Articles

Big optimization with genetic algorithms: Hadoop, Spark, and MPI

Solving problems of high dimensionality (and complexity) usually needs the intense use of technologies, like parallelism, advanced computers and new types of algorithms. MapReduce (MR) is a computing paradigm long time existing in computer science that has been proposed in the last years for dealing with big data applications, though it could also be used for many other tasks. In this article, we address big optimization: the solution to large instances of combinatorial optimization problems by using MR as the paradigm to design solvers that allow transparent runs on a varied number of computers that collaborate to find the problem solution. We study and analyze the MR technology, focusing on Hadoop, Spark, and MPI as the middleware platforms to develop genetic algorithms (GAs). From this, MRGA solvers arise using a different programming paradigm from the usual imperative transformational programming. Our objective is to confirm the expected benefits of these systems, namely file, memory, and communication management, over the resulting algorithms. We analyze our MRGA solvers from relevant points of view like scalability, speedup, and communication vs. computation time in big optimization. The results for high-dimensional datasets show that the MRGA over Hadoop outperforms the implementations in Spark and MPI frameworks. For the smallest datasets, the execution of MRGA on MPI is always faster than the executions of the remaining MRGAs. Finally, the MRGA over Spark presents the lowest communication times. Numerical and time insights are given in our work, so as to ease future comparisons of new algorithms over these three popular technologies.

Exploring search space trees using an adapted version of Monte Carlo tree search for combinatorial optimization problems

In this article we propose a heuristic algorithm to explore search space trees associated with instances of combinatorial optimization problems. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is used to explore game trees and represents the state-of-the-art algorithm for a number of games. Several enhancements to Monte Carlo tree search are proposed that make the algorithm more suitable in a combinatorial optimization context. These enhancements exploit the combinatorial structure of the problem and aim to efficiently explore the search space tree by pruning subtrees, using a heuristic simulation policy, reducing the domains of variables by eliminating dominated value assignments and using a beam width. The algorithm was implemented with its components specifically tailored to two combinatorial optimization problems: the quay crane scheduling problem with non-crossing constraints and the 0-1 knapsack problem. For the first problem our algorithm surpasses the state-of-the-art results and several new best solutions are found for a benchmark set of instances. For the second problem our algorithm typically produces near-optimal solutions that are slightly worse than the state-of-the-art results, but it needs only a small fraction of the time to do so. These results indicate that the algorithm is competitive with the state-of-the-art for two entirely different combinatorial optimization problems.

The algorithmic hardness threshold for continuous random energy models

We prove an algorithmic hardness result for finding low-energy states in the so-called continuous random energy model (CREM) , introduced by Bovier and Kurkova in 2004 as an extension of Derrida’s generalized random energy model . The CREM is a model of a randomenergy landscape (X_v)_{v \in \{0,1\}^N} on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of p -spin models including the Sherrington–Kirkpatrick model. The CREM is parameterized by an increasing function A \colon [0,1]\to[0,1] , which encodes the correlations between states. We exhibit an algorithmic hardness threshold x_* , which is explicit in terms of A . More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any \varepsilon > 0 a linear-time algorithm which finds states v \in \{0,1\}^N with X_v \ge (x_*-\varepsilon) N . Second, we show that the value x_* is essentially best-possible: for any \varepsilon > 0 , any algorithm which finds states v with X_v \ge (x_*+\varepsilon)N requires exponentially many queries in expectation and with high probability. We further discuss what insights this study yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

Growing Neural Gas as a Memory Mechanism of a Heuristic to Solve a Community Detection Problem in Networks

Iterative heuristics are commonly used to address combinatorial optimization problems. However, to meet both robustness and efficiency with these methods when their iterations are independent, it is necessary to consider a high number of iterations or to include local search-based strategies in them. Both approaches are very time-consuming and, consequently, not efficient for medium and large-scale instances of combinatorial optimization problems. In particular, the community detection problem in networks is well-known due to the instances with hundreds to thousands of vertices. In the literature, the heuristics to detect communities in networks that use a local search are those that achieve the partitions with the best solution values. Nevertheless, they are not suitable to tackle medium to large scale networks. This paper presents an adaptive heuristic, named GNGClus, that uses the neural network Growing Neural Gas to play the role of memory mechanism. The computational experiment with LFR networks indicates that the proposed strategy significantly outperformed the same solution method with no memory mechanism. In addition, GNGClus was very competitive with a version of the heuristic that employs an elite set of solutions to guide the solution search.

집합 커버링 문제를 위한 정수계획법 기반 지역 탐색

The set covering problem (SCP) is one of representative combinatorial optimization problems, which is defined as the problem of covering the m-rows by a subset of the n-columns at minimal cost. This paper proposes a method utilizing Integer Programming-based Local Search (IPbLS) to solve the set covering problem. IPbLS is a kind of local search technique in which the current solution is improved by searching neighborhood solutions. Integer programming is used to generate neighborhood solution in IPbLS. The effectiveness of the proposed algorithm has been tested on OR-Library test instances. The experimental results showed that IPbLS could search for the best known solutions in all the test instances. Especially, I confirmed that IPbLS could search for better solutions than the best known solutions in four test instances. ∙제1저자 : 황준하 ∙교신저자 : 황준하 ∙투고일 : 2014. 7. 22, 심사일 : 2014. 8. 11, 게재확정일 : 2014. 9. 11. * 금오공과대학교 컴퓨터공학과(Dept. of Computer Engineering, Kumoh National Institute of Technology) ※ 본 연구는 금오공과대학교 학술연구비에 의하여 연구된 논문임 14 Journal of The Korea Society of Computer and Information October 2014 ▸

FTH-B&B: A Fault-Tolerant HierarchicalBranch and Bound for Large ScaleUnreliable Environments

Solving to optimality large instances of combinatorial optimization problems using Brand and Bound (B&B) algorithms requires a huge amount of computing resources. In this paper, we investigate the design and implementation of such algorithms on computational grids. Most of existing grid-based B&B algorithms are based on the Master-Worker paradigm, their scalability is therefore limited. In addition, even if the volatility of resources is a major issue in grids fault tolerance is rarely addressed in these works. We thereby propose FTH-B&B, a fault tolerant hierarchical B&B. FTH-B&B is based on different new mechanisms enabling to efficiently build and maintain balanced the hierarchy, and to store and recover work units (sub-problems). FTH-B&B has been implemented on top of the ProActive grid middleware and programming environment and applied to the Flow-Shop scheduling problem. Very often, the validation of existing grid-based B&B works is performed either through simulation or a very small real grid. In this paper, we experimented FTH-B&B on the Grid’5000 real French nation-wide computational grid using up to 1,900 processor cores distributed over six sites. The reported results show that the overhead induced by the proposed mechanisms is very low and an efficiency close to 100 percent can be achieved on some Taillards benchmarks of the Flow-Shop problem. In addition, the results demonstrate the robustness of the proposed mechanisms even in extreme failure situations.

1A1-L01 JSPに対する部分的な割り付け規則の適用による局所探索法の提案(進化・学習とロボティクス)

Job-shop Scheduling Problem (JSP) is one of the representative combinatorial optimization problems where the objective is to find an ideal assignment of resources when multiple tasks must be processed simultaneously. Many heuristic methods for the JSP has been proposed. Among them, dispatching rules can provide a feasible schedule within a relatively short time. However, the obtained schedule is usually insufficient and further improvement by other heuristics is needed. Sub-problem is defined by assignments of resources within an arbitrary selected time window and rule-based reconstruction is performed by solving the sub-problem with a dispatching rule. An approprate set of time window and dispatching rule is obtained by genetic algorithm. This rule-based reconstruction procedure is combined with tabu search algorithm. The experiments are shows that the proposed algorithm performs better than an existing heuristic method when the number of tasks is much larger than the number of resources.

Solving the three dimensional quadratic assignment problem on a computational grid

The exact resolution of large instances of combinatorial optimization problems, such as three dimensional quadratic assignment problem (Q3AP), is a real challenge for grid computing. Indeed, it is necessary to reconsider the resolution algorithms and take into account the characteristics of such environments, especially large scale and dynamic availability of resources, and their multi-domain administration. In this paper, we revisit the design and implementation of the branch and bound algorithm for solving large combinatorial optimization problems such as Q3AP on the computational grids. Such gridification is based on new ways to efficiently deal with some crucial issues, mainly dynamic adaptive load balancing and fault tolerance. Our new approach allowed the exact resolution on a nation-wide grid of a difficult Q3AP instance. To solve this instance, an average of 1,123 computing cores were used for less than 12 days with a peak of around 3,427 computing cores.

On the one-dimensional stock cutting problem in the paper tube industry

The one-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems which arises in many industrial applications. Although the primary objective of 1D-CSP is to minimize the total length of used stock rolls, the efficiency of cutting processes has become more important in recent years. The crucial bottleneck of the cutting process often occurs at handling operations in semiautomated manufacturers such as those in the paper tube industry. To reduce interruptions and errors at handling operations in the paper tube industry, we consider a variant of 1D-CSP that minimizes the total length of used stock rolls while constraining (C1) the number of setups of each stock roll type, (C2) the combination of piece lengths occurring in open stacks simultaneously, and (C3) the number of open stacks. For this problem, we propose a generalization of the cutting pattern called the "cutting group," which is a sequence of cutting patterns that satisfies the given upper bounds of setups of each stock roll type and open stacks. To generate good cutting groups, we decompose the 1D-CSP into a number of auxiliary bin packing problems. We develop a tabu search algorithm based on a shift neighborhood that solves the auxiliary bin packing problems by the first-fit decreasing heuristic algorithm. Experimental results show that our algorithm improves the quality of solutions compared to the existing algorithm used in a paper tube factory.

RELAXATION HEURISTICS FOR THE SET COVERING PROBLEM(<Special Issue>the 50th Anniversary of the Operations Research Society of Japan)

The set covering problem (SCP) is one of representative combinatorial optimization problems, which has many practical applications. The continuous development of mathematical programming has derived a number of impressive heuristic algorithms as well as exact branch-and-bound algorithms, which can solve huge SCP instances of bus, railway and airline crew scheduling problems. We survey heuristic algorithms for SCP focusing mainly on contributions of mathematical programming techniques to heuristics, and illustrate their performance through experimental analysis.

One-Dimensional Cutting Stock Problem with a Given Number of Setups: A Hybrid Approach of Metaheuristics and Linear Programming

One-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems, which arises in many industrial applications. Since the setup costs for switching different cutting patterns become more dominant in recent cutting industry, we consider a variant of 1D-CSP, called the pattern restricted problem (PRP), to minimize the number of stock rolls while constraining the number of different cutting patterns within a bound given by users. For this problem, we propose a local search algorithm that alternately uses two types of local search processes with the 1-add neighborhood and the shift neighborhood, respectively. To improve the performance of local search, we incorporate it with linear programming (LP) techniques, to reduce the number of solutions in each neighborhood. A sensitivity analysis technique is introduced to solve a large number of associated LP problems quickly. Through computational experiments, we observe that the new algorithm obtains solutions of better quality than those obtained by other existing approaches.

Very Large-Scale Neighborhood Search for the K-Constraint Multiple Knapsack Problem

The K-Constraint Multiple Knapsack Problem (K-MKP) is a generalization of the multiple knapsack problem, which is one of the representative combinatorial optimization problems known to be NP-hard. In K-MKP, each item has K types of weights and each knapsack has K types of capacity. In this paper, we propose several very large-scale neighborhood search (VLSN) algorithms to solve K-MKP. One of the VLSN algorithms incorporates a novel approach that consists of randomly perturbing the current solution in order to efficiently produce a set of simultaneous non-profitable moves. These moves would allow several items to be transferred from their current knapsacks and assigned to new knapsacks, which makes room for new items to be inserted through multi-exchange movements and allows for improved solutions. Computational results presented show that the method is effective, and provides better solutions compared to exact algorithms run for the same amount of time.

A very large-scale neighborhood search algorithm for the multi-resource generalized assignment problem

We propose a metaheuristic algorithm for the multi-resource generalized assignment problem (MRGAP). MRGAP is a generalization of the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features a very large-scale neighborhood search, which is a mechanism of conducting the search with complex and powerful moves, where the resulting neighborhood is efficiently searched via the improvement graph. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational comparisons on benchmark instances show that the method is effective, especially for types D and E instances, which are known to be quite difficult.

An Ejection Chain Approach for the Generalized Assignment Problem

We propose a tabu search algorithm for the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features an ejection chain approach, which is embedded in a neighborhood construction to create more complex and powerful moves. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational results on benchmark instances of small sizes show that the method obtains solutions that are optimal or that deviate by at most 0.16% from the best known solutions. Comparisons with other approaches from the literature show that, for instances of larger sizes, our method obtains the best solutions among all heuristics tested.

A variable depth search algorithm with branching search for the generalized assignment problem

In this paper, we propose a variable depth search (VDS) algorithm for the generalized assignment problem (GAP), which is one of the representative combinatorial optimization problems, and is known to be NP-hard. The VDS is a generalization of the local search. The main idea of VDS is to change the size of the neighborhood adaptively so that the algorithm can effectively traverse larger search space within reasonable computational time. In our previous paper (M. Yagiura, T. Yamaguchi and T. Ibaraki, “A variable depth search algorithm for the generalized assignment problem,” Proc. 2nd Metaheuristics International Conference (MIC97), 1997 129-130 (full version is to appear in the post-conference book)), we proposed a simple VDS algorithm for the GAP, and obtained good results. To further improve the performance of the VDS, we examine the effectiveness of incorporating branching search processes to construct the neighborhoods. Various types of branching rules are examined, and it is observed that appropriate choices of branching strategies improve the performance of VDS. Comparisons with other existing heuristics are also conducted using benchmark instances. The proposed algorithm is found to be quite effective.

The use of dynamic programming in genetic algorithms for permutation problems

To deal with computationally hard problems, approximate algorithms are used to provide reasonably good solutions in practical time. Genetic algorithms are an example of the meta-heuristics which were recently introduced and which have been successfully applied to a variety of problems. We propose to use dynamic programming in the process of obtaining new generation solutions in the genetic algorithm, and call it a genetic DP algorithm. To evaluate the effectiveness of this approach, we choose three representative combinatorial optimization problems, the single machine scheduling problem, the optimal linear arrangement problem and the traveling salesman problem, all of which ask to compute optimum permutations of n objects and are known to be NP-hard. Computational results for randomly generated problem instances exhibit encouraging features of genetic DP algorithms.