A parallel maximum clique algorithm for large and massive sparse graphs

This paper describes BBMCPara, a new parallel exact maximum clique algorithm tailored for large and massive sparse graphs. The paper first presents a sequential algorithm BBMCSP, which builds on ideas from a leading bit-parallel published algorithm for middle-size graphs. It employs heavy pre-processing and a new sparse bitset encoding to outperform other state-of-the-art algorithms by up to several orders of magnitude over a set of real networks. BBMCPara parallelizes BBMCSP by splitting according to a preprocessing step of the latter. On a 20-core computer, it averages speedups close to an order of magnitude over real graphs of up to 3 million vertices. According to the reported results, BBMCPara appears to be the current fastest algorithm for large and massive real networks to the best of our knowledge.     

A hybrid scatter search/electromagnetism meta-heuristic for project scheduling

In the last few decades, several effective algorithms for solving the resource-constrained project scheduling problem have been proposed. However, the challenging nature of this problem, summarised in its strongly NP-hard status, restricts the effectiveness of exact optimisation to relatively small instances. In this paper, we present a new meta-heuristic for this problem, able to provide near-optimal heuristic solutions for relatively large instances. The procedure combines elements from scatter search, a generic population-based evolutionary search method, and from a recently introduced heuristic method for the optimisation of unconstrained continuous functions based on an analogy with electromagnetism theory. We present computational experiments on standard benchmark datasets, compare the results with current state-of-the-art heuristics, and show that the procedure is capable of producing consistently good results for challenging instances of the resource-constrained project scheduling problem. We also demonstrate that the algorithm outperforms state-of-the-art existing heuristics.     

Graph 3-coloring with a hybrid self-adaptive evolutionary algorithm

This paper proposes a hybrid self-adaptive evolutionary algorithm for graph coloring that is hybridized with the following novel elements: heuristic genotype-phenotype mapping, a swap local search heuristic, and a neutral survivor selection operator. This algorithm was compared with the evolutionary algorithm with the SAW method of Eiben et al., the Tabucol algorithm of Hertz and de Werra, and the hybrid evolutionary algorithm of Galinier and Hao. The performance of these algorithms were tested on a test suite consisting of randomly generated 3-colorable graphs of various structural features, such as graph size, type, edge density, and variability in sizes of color classes. Furthermore, the test graphs were generated including the phase transition where the graphs are hard to color. The purpose of the extensive experimental work was threefold: to investigate the behavior of the tested algorithms in the phase transition, to identify what impact hybridization with the DSatur traditional heuristic has on the evolutionary algorithm, and to show how graph structural features influence the performance of the graph-coloring algorithms. The results indicate that the performance of the hybrid self-adaptive evolutionary algorithm is comparable with, or better than, the performance of the hybrid evolutionary algorithm which is one of the best graph-coloring algorithms today. Moreover, the fact that all the considered algorithms performed poorly on flat graphs confirms that graphs of this type are really the hardest to color.     

A Combined Evolutionary Search and Multilevel Optimisation Approach to Graph-Partitioning

The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages.     

Hybrid evolutionary optimization of the operation of pipeless plants

Pipeless plants are a new production concept in chemical engineering in which automated guided vehicles (AGVs) transport the substances in mobile vessels between processing stations. In the operation of such plants, decisions have to be made on the scheduling of the production, the assignment of the equipment and the routing of the AGVs that carry the vessels. The large number of interacting degrees of freedom prohibit the use of exact mathematical algorithms to compute optimal schedules. This paper describes the combination of an evolutionary scheduling algorithm with a simulation based schedule builder. The algorithm is tested on a real-life example and on a benchmark problem from the literature and yields considerably shorter makespans than a heuristic solution.     

Tutorial on Surrogate Constraint Approaches for Optimization in Graphs

Surrogate constraint methods have been embedded in a variety of mathematical programming applications over the past thirty years, yet their potential uses and underlying principles remain incompletely understood by a large segment of the optimization community. In a number of significant domains of combinatorial optimization, researchers have produced solution strategies without recognizing that they can be derived as special instances of surrogate constraint methods. Once the connection to surrogate constraint ideas is exposed, additional ways to exploit this framework become visible, frequently offering opportunities for improvement.We provide a tutorial on surrogate constraint approaches for optimization in graphs, illustrating the key ideas by reference to independent set and graph coloring problems, including constructions for weighted independent sets which have applications to associated covering and weighted maximum clique problems. In these settings, the surrogate constraints can be generated relative to well-known packing and covering formulations that are convenient for exposing key notions. The surrogate constraint approaches yield widely used heuristics for identifying independent sets as simple special cases, and also afford previously unidentified heuristics that have greater power in these settings. Our tutorial also shows how the use of surrogate constraints can be placed within the context of vocabulary building strategies for independent set and coloring problems, providing a framework for applying surrogate constraints that can be used in other applications.At a higher level, we show how to make use of surrogate constraint information, together with specialized algorithms for solving associated sub-problems, to obtain stronger objective function bounds and improved choice rules for heuristic or exact methods. The theorems that support these developments yield further strategies for exploiting surrogate constraint relaxations, both in graph optimization and integer programming generally.     

General Variable Neighborhood Search for computing graph separators

Computing graph separators in networks has a wide range of real-world applications. For instance, in telecommunication networks, a separator determines the capacity and brittleness of the network. In the field of graph algorithms, the computation of balanced small-sized separators is very useful, especially for divide-and-conquer algorithms. In bioinformatics and computational biology, separators are required in grid graphs providing a simplified representation of proteins. This papers presents a new heuristic algorithm based on the Variable Neighborhood Search methodology for computing vertex separators. We compare our procedure with the state-of-the-art methods. Computational results show that our procedure obtains the optimum solution in all of the small and medium instances, and high-quality results in large instances.     

Solving multiobjective,multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive ε-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive ε -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).     

Generating Hard Test Instances with Known Optimal Solution for the Rectilinear Quadratic Assignment Problem

In this paper we consider the rectilinear version of the quadratic assignment problem (QAP). We define a class of edge-weighted graphs with nonnegatively valued bisections. For one important type of such graphs we provide a characterization of point sets on the plane for which the optimal value of the related QAP is zero. These graphs are used in the algorithms for generating rectilinear QAP instances with known provably optimal solutions. The basic algorithm of such type uses only triangles. Making a reduction from 3-dimensional matching, it is shown that the set of instances which can be generated by this algorithm is hard. The basic algorithm is extended to process graphs larger than triangles. We give implementation details of this extension and of four other variations of the basic algorithm. We compare these five and also two existing generators experimentally employing multi-start descent heuristic for the QAP as an examiner. The graphs with nonnegatively valued bisections can also be used in the construction of lower bounds on the optimal value for the rectilinear QAP.     

Length of clustering algorithms based on random walks with an application to neuroscience

In this paper we show how the notions of conductance and cutoff can be used to determine the length of the random walks in some clustering algorithms. We consider graphs which are globally sparse but locally dense. They present a community structure: there exists a partition of the set of vertices into subsets which display strong internal connections but few links between each other. Using a distance between nodes built on random walks we consider a hierarchical clustering algorithm which provides a most appropriate partition. The length of these random walks has to be chosen in advance and has to be appropriate. Finally, we introduce an extension of this clustering algorithm to dynamical sequences of graphs on the same set of vertices.     

The static stochastic knapsack problem with normally distributed item sizes

This paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack. An item’s value is given by the realization of the product of a random unit revenue and the random item size. When the realization of the sum of selected item sizes exceeds the knapsack capacity, a penalty cost is incurred for each unit of overflow, while our model allows for a salvage value for each unit of capacity that remains unused. We seek to maximize the expected net profit resulting from the assignment of items to the knapsack. Although the capacity is fixed in our core model, we show that problems with random capacity, as well as problems in which capacity is a decision variable subject to unit costs, fall within this class of problems as well. We focus on the case where item sizes are independent and normally distributed random variables, and provide an exact solution method for a continuous relaxation of the problem. We show that an optimal solution to this relaxation exists containing no more than two fractionally selected items, and develop a customized branch-and-bound algorithm for obtaining an optimal binary solution. In addition, we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high-quality solutions.     

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.     

Computing inertia sets using atoms

We consider the problem of computing inertia sets for graphs. By using tools for combining the inertia sets of smaller graphs we can reduce this problem to understanding the inertia sets for three-connected graphs that are not joins. We term such graphs atoms and give the inertia sets for all atoms on at most seven vertices. This can be used to compute the inertia sets for all graphs on at most seven vertices.     

Applying local search to the feedback vertex set problem

The feedback vertex set problem (FVSP) consists in making a given directed graph acyclic by removing as few vertices as possible. In spite of the importance of this NP-hard problem, no local search approach had been proposed so far for tackling it. Building on a property of acyclic graphs, we suggest in this paper a new representation of the solutions of the FVSP (feedback sets). Thanks to this solution representation, we are able to design a local transformation (equivalent to a neighborhood) that changes a feedback set into a new one. Based on this neighborhood, we have developed a simulated annealing algorithm for the FVSP. Our experiments show that our algorithm outperforms the best existing heuristic, namely the greedy adaptive search procedure by Pardalos et al.     

An evolutionary approach for bandwidth multicoloring problems

In this paper we consider some generalizations of the vertex coloring problem, where distance constraints are imposed between adjacent vertices (bandwidth coloring problem) and each vertex has to be colored with more than one color (bandwidth multicoloring problem). We propose an evolutionary metaheuristic approach for the first problem, combining an effective tabu search algorithm with population management procedures. The approach can be applied to the second problem as well, after a simple transformation. Computational results on instances from the literature show that the overall algorithm is able to produce high quality solutions in a reasonable amount of time, outperforming the most effective algorithms proposed for the bandwidth coloring problem, and improving the best known solution of many instances of the bandwidth multicoloring problem.     

A computational study of several heuristics for the DRPP

The problem of designing a route of minimum length for a postman that starts and finishes at his office and has to deliver the mail along a set of streets in a city is known as the Rural Postman Problem. When the postman has to obey the directions of the streets, we have the directed version of this problem. Finding an exact solution, in the general case, is intractably difficult. Hence, we have implemented three heuristic algorithms for approximately solving this problem and a procedure for obtaining a lower bound to the optimal length. Also, we present numerical experimentations based on a collection of random instances with up to 30 connected components, 240 vertices and 801 arcs. A lower bound for the optimal DRPP value is obtained by using cutting plane techniques, producing an optimal solution in 21 out of 60 instances. The main purpose of this work is to compare these three algorithms. We also give guidelines concerning the performance of the algorithms depending on the characteristics of the problem to solve.     

Exact weighted vertex coloring via branch-and-price

We consider the Weighted Vertex Coloring Problem (WVCP), in which a positive weight is associated to each vertex of a graph. In WVCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used, where the cost of each color is given by the maximum weight of the vertices assigned to that color. This NP-hard problem arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose the first exact algorithm for the problem, which is based on column generation and branch-and-price. Computational results on a large set of instances from the literature are reported, showing excellent performance when compared with the best heuristic algorithms from the literature.     

A branch-and-cut algorithm based on semidefinite programming for the minimum <Emphasis Type="Italic">k</Emphasis>-partition problem

The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k≥3.     

Memetic algorithm for the antibandwidth maximization problem

The antibandwidth maximization problem (AMP) consists of labeling the vertices of a n-vertex graph G with distinct integers from 1 to n such that the minimum difference of labels of adjacent vertices is maximized. This problem can be formulated as a dual problem to the well known bandwidth problem. Exact results have been proved for some standard graphs like paths, cycles, 2 and 3-dimensional meshes, tori, some special trees etc., however, no algorithm has been proposed for the general graphs. In this paper, we propose a memetic algorithm for the antibandwidth maximization problem, wherein we explore various breadth first search generated level structures of a graph—an imperative feature of our algorithm. We design a new heuristic which exploits these level structures to label the vertices of the graph. The algorithm is able to achieve the exact antibandwidth for the standard graphs as mentioned. Moreover, we conjecture the antibandwidth of some 3-dimensional meshes and complement of power graphs, supported by our experimental results.