An effective multilevel tabu search approach for balanced graph partitioning

Graph partitioning is one of the fundamental NP-complete problems which is widely applied in many domains, such as VLSI design, image segmentation, data mining, etc. Given a graph G=(V,E), the balanced k-partitioning problem consists in partitioning the vertex set V into k disjoint subsets of about the same size, such that the number of cutting edges is minimized. In this paper, we present a multilevel algorithm for balanced partition, which integrates a powerful refinement procedure based on tabu search with periodic perturbations. Experimental evaluations on a wide collection of benchmark graphs show that the proposed approach not only competes very favorably with the two well-known partitioning packages METIS and CHACO, but also improves more than two thirds of the best balanced partitions ever reported in the literature.     

A method of finding simple disjunctive decomposition with one free variable

Simple disjunctive decomposition with one free variable can be obtained by using the prime implicants of the Boolean function. A function is decomposable with its redundant variable as a free variable. A necessary condition for decomposition with a non-redundant variable as free variable is that the number of minterms in an n variable function be 2^(n?1). A sufficient condition is that the variable remain present, in either true or complemented form, in all the prime implicants of the function.     

Shipment partitioning and routing to minimize makespan in a transportation network

This paper addresses the problem of partitioning and transporting a shipment of known size through an n-node public transportation network with known scheduled departure and arrival times and expected available capacities for each departure. The objective is to minimize the makespan of shipping. The problem while practical in its scope, has received very little attention in the literature perhaps because of the concentration of research in vehicle routing without regard to partitioning and partitioning without regard to routing. A general non-linear programming model is developed. The model is then converted into a linear model through the Routing First and Assignment Second approach. This approach is different from the general decomposition approaches since they normally do not guarantee optimality. However, the linear model still involves a large number of constraints, and solution is not attempted here. Instead, three heuristics are proposed for solving the problem. Two of the heuristics use iterative techniques to evaluate all possible paths. The third heuristic uses a max-flows approach based upon aggregated capacities to reduce the size of the network presented to the other heuristics. This allows for a good starting point for other heuristics, and may impact the total computational effort. We find that the heuristics developed perform well because in the case of networks that are not congested, they find the optimal solution.     

A new approach to the decomposition of boolean functions. 4. non-disjoint decomposition: the method of <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>-partitions

A set-theoretical approach to the non-disjoint decomposition of different forms of representation of Boolean functions of n variables is considered. This approach is based on the method of p,q-partitions of conjuncterms and on the concept of a decomposition clone. Two techniques of searching for some non-disjoint functional decomposition are described. Theorems on the non-disjoint decomposition of full and partial functions and their systems are formulated. The proposed approach is illustrated by examples. Parts I, II, and III of this article are published in No. 5 (2001), No. 1 (2002), and No. 2 (2007). Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 15-41, May–June 2009.     

A graph coloring heuristic using partial solutions and a reactive tabu scheme

Most of the recent heuristics for the graph coloring problem start from an infeasible k-coloring (adjacent vertices may have the same color) and try to make the solution feasible through a sequence of color exchanges. In contrast, our approach (called FOO-PARTIALCOL), which is based on tabu search, considers feasible but partial solutions and tries to increase the size of the current partial solution. A solution consists of k disjoint stable sets (and, therefore, is a feasible, partial k-coloring) and a set of uncolored vertices. We introduce a reactive tabu tenure which substantially enhances the performance of both our heuristic as well as the classical tabu algorithm (called TABUCOL) proposed by Hertz and de Werra [Using tabu search techniques for graph coloring, Computing 1987;39:345–51]. We will report numerical results on different benchmark graphs and we will observe that FOO-PARTIALCOL, though very simple, outperforms TABUCOL on some instances, provides very competitive results on a set of benchmark graphs which are known to be difficult, and outperforms the best-known methods on the graph flat300_28_0. For this graph, FOO-PARTIALCOL finds an optimal coloring with 28 colors. The best coloring achieved to date uses 31 colors. Algorithms very close to TABUCOL are still used as intensification procedures in the best coloring methods, which are evolutionary heuristics. FOO-PARTIALCOL could then be a powerful alternative. In conclusion FOO-PARTIALCOL is one of the most efficient simple local search coloring methods yet available.     

Finding optimal solutions to the graph partitioning problem with heuristic search

As search spaces become larger and as problems scale up, an efficient way to speed up the search is to use a more accurate heuristic function. A better heuristic function might be obtained by the following general idea. Many problems can be divided into a set of subproblems and subgoals that should be achieved. Interactions and conflicts between unsolved subgoals of the problem might provide useful knowledge which could be used to construct an informed heuristic function. In this paper we demonstrate this idea on the graph partitioning problem (GPP). We first show how to format GPP as a search problem and then introduce a sequence of admissible heuristic functions estimating the size of the optimal partition by looking into different interactions between vertices of the graph. We then optimally solve GPP with these heuristics. Experimental results show that our advanced heuristics achieve a speedup of up to a number of orders of magnitude. Finally, we experimentally compare our approach to other states of the art graph partitioning optimal solvers on a number of classes of graphs. The results obtained show that our algorithm outperforms them in many cases.     

Compact DSOP and Partial DSOP Forms

Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP) of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals f, such that no two products cover the same minterm of f. DSOP forms are a special instance of partial DSOPs, i.e. the general case where a subset of minterms must be covered exactly once and the other minterms (typically corresponding to don’t care conditions of f) can be covered any number of times. We discuss finding DSOPs and partial DSOPs with a minimal number of products, a problem theoretically connected with various properties of Boolean functions and practically relevant in the synthesis of digital circuits. Finding an absolute minimum is hard, in fact we prove that the problem of absolute minimization of partial DSOPs is NP-hard. Therefore it is crucial to devise a polynomial time heuristic that compares favorably with the known minimization tools. To this end we develop a further piece of theory starting from the definition of the weight of a cube c as a functions of the number of fragments induced on other cubes by the selection of c, and show how cube weights can be exploited for building a class of minimization heuristics for DSOP and partial DSOP synthesis. A set of experiments conducted on major benchmark functions show that our method, with a family of variants, always generates better results than the ones of previous heuristics, including the method based on a BDD representation of f.     

Clustering search algorithm for the capacitated centered clustering problem

The capacitated centered clustering problem (CCCP) consists in partitioning a set of n points into p disjoint clusters with a known capacity. Each cluster is specified by a centroid. The objective is to minimize the total dissimilarity within each cluster, such that a given capacity limit of the cluster is not exceeded. This paper presents a solution procedure for the CCCP, using the hybrid metaheuristic clustering search (CS), whose main idea is to identify promising areas of the search space by generating solutions through a metaheuristic and clustering them into groups that are then further explored with local search heuristics. Computational results in test problems of the literature show that the CS found a significant number of new best-known solutions in reasonable computational times.     

Mixing Static and Dynamic Partitioning to Parallelize a Constraint Programming Solver

This paper presents an external parallelization of Constraint Programming (CP) search tree mixing both static and dynamic partitioning. The principle of the parallelization is to partition the CP search tree into a set of sub-trees, then assign each sub-tree to one computing core in order to perform a local search using a sequential CP solver. In this context, static partitioning consists of decomposing the CP variables domains in order to split the CP search tree into a set of disjoint sub-trees to assign them to the cores. This strategy performs well without adding an extra cost to the parallel search, but the problem is the load imbalance between computing cores. On the other hand, dynamic partitioning is based on preservation of the search state to generate, dynamically or on demand, the sub-trees that are assigned to the cores. This strategy offers good load balancing between the different computing cores, but computing overcosts appear due to the initialisation of the search when a sub-tree is migrated from one core to another. In this paper, we propose a new partitioning strategy that mixes the static and dynamic partitioning and enjoys the benefits of each strategy. This mixed partitioning is designed to run on shared and distributed memory architectures. The performances obtained are illustrated by solving the CP problems modelled using the FlatZinc format and solved using the Google OR-Tools solver on top of the parallel Bobpp framework.     

Unbiased split selection for classification trees based on the Gini Index

Classification trees are a popular tool in applied statistics because their heuristic search approach based on impurity reduction is easy to understand and the interpretation of the output is straightforward. However, all standard algorithms suffer from a major problem: variable selection based on standard impurity measures as the Gini Index is biased. The bias is such that, e.g., splitting variables with a high amount of missing values—even if missing completely at random (MCAR)—are artificially preferred. A new split selection criterion that avoids variable selection bias is introduced. The exact distribution of the maximally selected Gini gain is derived by means of a combinatorial approach and the resulting p-value is suggested as an unbiased split selection criterion in recursive partitioning algorithms. The efficiency of the method is demonstrated in simulation studies and a real data study from veterinary gynecology in the context of binary classification and continuous predictor variables with different numbers of missing values. The proposed method is extendible to categorical and ordinal predictor variables and to other split selection criteria such as the cross-entropy.     

Local search heuristics for the mobile facility location problem

In the mobile facility location problem (MFLP), one seeks to relocate (or move) a set of existing facilities and assign clients to these facilities so that the sum of facility movement costs and the client travel costs (each to its assigned facility) is minimized. This paper studies formulations and develops local search heuristics for the MFLP. First, we develop an integer programming (IP) formulation for the MFLP by observing that for a given set of facility destinations the problem may be decomposed into two polynomially solvable subproblems. This IP formulation is quite compact in terms of the number of nonzero coefficients in the constraint matrix and the number of integer variables; and allows for the solution of large-scale MFLP instances. Using the decomposition observation, we propose two local search neighborhoods for the MFLP. We report on extensive computational tests of the new IP formulation and local search heuristics on a large range of instances. These tests demonstrate that the proposed formulation and local search heuristics significantly outperform the existing formulation and a previously developed local search heuristic for the problem.     

A New Approach to the Decomposition of Boolean Functions by the Method of q-Partitions. II. Repeated Decomposition

A new approach to the decomposition of Boolean functions that depend on n variables and are represented in various forms is considered. The approach is based on the method of #-partitioning of minterms and on the introduced concept of a decomposition clone. The theorem on simple disjunctive decomposition of full and partial functions is formulated. The approach proposed is illustrated by examples.     

Optimal complexity reduction of polyhedral piecewise affine systems

This paper focuses on the NP-hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude.     

A hybrid heuristic for the overlapping cluster editing problem

Clustering problems are applicable to several areas of science. Approaches and algorithms are as varied as the applications. From a graph theory perspective, clustering can be generated by partitioning an input graph into a vertex-disjoint union of cliques (clusters) through addition and deletion of edges. Finding the minimum number of edges additions and deletions required to cluster data that can be represented as graphs is a well-known problem in combinatorial optimization, often referred to as cluster editing problem. However, many real-world clustering applications are characterized by overlapping clusters, that is, clusters that are non-disjoint. In these situations cluster editing cannot be applied to these problems. Literature concerning a relaxation of the cluster editing, where clusters can overlap, is scarce. In this work, we propose the overlapping cluster editing problem, a variation of the cluster editing where the goal is to partition a graph, also by editing edges, into maximal cliques that are not necessarily disjoint. In addition, we also present three slightly different versions of a hybrid heuristic to solve this problem. Each hybrid heuristic is based on coupling two metaheuristicsthat, together, generate a set of clusters; and one of three mixed-integer linear programming models, also introduced in this paper, that uses these clusters as input. The objective with the metaheuristics is to limit the solution exploration space in the models’ resolution, therefore reducing its computational time.Tests results show that the all proposed hybrid heuristic versions are able to generate good-quality overlapping cluster editing solutions. In particular, one version of the hybrid heuristic achieved, at a low computational cost, the best results in 51 of 112 randomly-generated graphs. Although the other two hybrid heuristic versions have harder to solve models, they obtained reasonable results in medium-sized randomly-generated graphs. In addition, the hybrid heuristic achieved good results identifying labeled overlapping clusters in a supervised data set experiment. Furthermore, we also show that, with our new problem definition, clustering a vertex in more than one cluster can reduce the edges editing cost.     

A New Approach to the Decomposition of Boolean Functions by the Method of q-Partitions. I. Separating Decomposition of Full and Partial Functions

A new approach to the decomposition of Boolean functions of n variables is considered; the functions being decomposed can be represented in various forms. The approach is based on the method of q-partitions of minterms and on the introduced concept of a decomposition clone. The theorem on simple separating decomposition of full and partial functions is formulated. The approach proposed is illustrated by examples.     

An effective heuristic algorithm for sum coloring of graphs

Given an undirected graph G=(V,E), the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of G, using colors represented by natural numbers (1,2,…) such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present EXSCOL, a heuristic algorithm based on independent set extraction for this NP-hard problem. EXSCOL identifies iteratively collections of disjoint independent sets of equal size and assign to each independent set the smallest available color. For the purpose of computing large independent sets, EXSCOL employs a tabu search based heuristic. Experimental evaluations on a collection of 52 DIMACS and COLOR2 benchmark graphs show that the proposed approach achieves highly competitive results. For more than half of the graphs used in the literature, our approach improves the current best known upper bounds.     

An experimental evaluation of local search heuristics for graph partitioning

A common problem that arises in many applications is to partition the vertices of a graph intok subsets, each containing a bounded number of vertices, such that the number of graph edges with endpoints in different subsets is minimized. This paper describes an empirical study of the performance of various local search heuristics for thisk-way graph partitioning problem. The heuristics examined are local optimization, simulated annealing, tabu search, and genetic algorithms. In addition, the hierarchical hybrid approach is introduced, in which the problem is recursively decomposed into small pieces, to which local search heuristics are then applied.     

Partition-Based Lower Bound for Max-CSP

In this paper we address Max-CSP, a constraint optimization problem typically solved using a branch and bound scheme. It is well known that the efficiency of branch and bound greatly depends on the quality of the available lower bound. Previous approaches aggregate to the lower bound individual contributions of unassigned variables. In this paper, we augment this approach by adding global contributions of disjoint subsets of unassigned variables, which requires a partition of the set of unassigned variables. Using this idea, we introduce the partition-based lower bound. It improves previous approaches based on individual contributions in the sense that our method can be added up to previous bounds, possibly increasing their value. We demonstrate our method presenting two new algorithms, PFC-PRDAC and PFC-MPRDAC, which are the natural successors of PFC-RDAC and PFC-MRDAC augmented with our approach. We provide experimental evidence for the superiority of the new algorithms on random problems and real instances of weighted over-constrained problems.     

Hierarchical minimization of completion time variance and makespan in jobshops

This paper addresses the problem of minimizing makespan for a given set of n jobs to be processed on each of m machines in a static jobshop, subject to the minimum completion time variance (CTV). A lower bound on CTV is developed for the static jobshop problem. A backward scheduling approach is proposed using the observations on the development of lower bound for hierarchical minimization of CTV and makespan. A lower bound on makespan subject to minimum CTV is also presented for this problem. Finally, we present two simulated annealing heuristic approaches using the concepts of forward and backward scheduling. Their performances are compared against each other through the use of the lower bounds established in this work. The simulated annealing heuristic based on backward scheduling is shown to perform well by evaluating the developed heuristics on 82 jobshop problems taken from literature.     

Parameter tuning of a choice-function based hyperheuristic using Particle Swarm Optimization

A Constraint Satisfaction Problem is defined by a set of variables and a set of constraints, each variable has a nonempty domain of possible values. Each constraint involves some subset of the variables and specifies the allowable combinations of values for that subset. A solution of the problem is defined by an assignment of values to some or all of the variables that does not violate any constraints. To solve an instance, a search tree is created and each node in the tree represents a variable of the instance. The order in which the variables are selected for instantiation changes the form of the search tree and affects the cost of finding a solution. In this paper we explore the use of a Choice Function to dynamically select from a set of variable ordering heuristics the one that best matches the current problem state in order to show an acceptable performance over a wide range of instances. The Choice Function is defined as a weighted sum of process indicators expressing the recent improvement produced by the heuristic recently used. The weights are determined by a Particle Swarm Optimization algorithm in a multilevel approach. We report results where our combination of strategies outperforms the use of individual strategies.