Heuristics and metaheuristics for the maximum diversity problem

This paper presents extensive computational experiments to compare 10 heuristics and 20 metaheuristics for the maximum diversity problem (MDP). This problem consists of selecting a subset of maximum diversity from a given set of elements. It arises in a wide range of real-world settings and we can find a large number of studies, in which heuristic and metaheuristic methods are proposed. However, probably due to the fact that this problem has been referenced under different names, we have only found limited comparisons with a few methods on some sets of instances. This paper reviews all the heuristics and metaheuristics for finding near-optimal solutions for the MDP. We present the new benchmark library MDPLIB, which includes most instances previously used for this problem, as well as new ones, giving a total of 315. We also present an exhaustive computational comparison of the 30 methods on the MDPLIB. Non-parametric statistical tests are reported in our study to draw significant conclusions.     

Cooperating local search for the maximum clique problem

The advent of desktop multi-core computers has dramatically improved the usability of parallel algorithms which, in the past, have required specialised hardware. This paper introduces cooperating local search (CLS), a parallelised hyper-heuristic for the maximum clique problem. CLS utilises cooperating low level heuristics which alternate between sequences of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, where vertices of the current clique are swapped with vertices not in the current clique. These low level heuristics differ primarily in their vertex selection techniques and their approach to dealing with plateaus. To improve the performance of CLS, guidance information is passed between low level heuristics directing them to particular areas of the search domain. In addition, CLS dynamically reconfigures the allocation of low level heuristics to cores, based on information obtained during a trial, to ensure that the mix of low level heuristics is appropriate for the instance being optimised. CLS has no problem instance dependent parameters, improves the state-of-the-art performance for the maximum clique problem over all the BHOSLIB benchmark instances and attains unprecedented consistency over the state-of-the-art on the DIMACS benchmark instances.     

Tabu search and GRASP for the maximum diversity problem

In this paper, we develop new heuristic procedures for the maximum diversity problem (MDP). This NP-hard problem has a significant number of practical applications such as environmental balance, telecommunication services or genetic engineering. The proposed algorithm is based on the tabu search methodology and incorporates memory structures for both construction and improvement. Although proposed in seminal tabu search papers, memory-based constructions have often been implemented in naïve ways that disregard important elements of the fundamental tabu search proposals. We will compare our tabu search construction with a memory-less design and with previous algorithms recently developed for this problem. The constructive method can be coupled with a local search procedure or a short-term tabu search for improved outcomes. Extensive computational experiments with medium and large instances show that the proposed procedure outperforms the best heuristics reported in the literature within short computational times.     

Fast local search for the maximum independent set problem

Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.     

Subgraph extraction and metaheuristics for the maximum clique problem

The maximum clique problem involves finding the largest set of pairwise adjacent vertices in a graph. The problem is classic but still attracts much attention because of its hardness and its prominent applications. Our work is based on the existence of an order of all the vertices whereby those belonging to a maximum clique stay close enough to each other. Such an order can be identified via the extraction of a particular subgraph from the original graph. The problem can consequently be seen as a permutation problem that can be addressed efficiently by metaheuristics. We first design a memetic algorithm (MA) for this purpose. Computational experiments conducted on the DIMACS benchmark instances clearly show that our MA not only outperforms existing genetic approaches, but it also compares very well to state-of-the-art algorithms regarding the maximal clique size found after different runs. Furthermore, we show that a hybridization of MA with an iterated local search (ILS) improves the stability of the algorithm. This hybridization (MA-ILS) permits to find two distinct maximal cliques of size 79 and one of size 80 for the C2000.9 instance of the DIMACS benchmark.     

A two-phase tabu search based evolutionary algorithm for the maximum diversity problem

In this paper, we study the maximum diversity problem (MDP) which is equivalent to the quadratic unconstrained binary optimization (QUBO) problem with cardinality constraint. The MDP aims to select a subset of elements with given cardinality such that the sum of pairwise distances between any two elements in the selected subset is maximized. For solving this computationally challenging problem, we propose a two-phase tabu search based evolutionary algorithm (TPTS/EA), which integrates several distinguishing features to ensure the diversity and the quality of the evolution, such as a two-phase tabu search algorithm which consists of a dynamic candidate list (DCL) strategy-based traditional tabu search in the first phase and a solution-based tabu search procedure to refine the search in the second phase, and two path-relinking based recombination operators to generate new offspring solutions. Tested on three sets of totally 140 public instances in the literature, the study demonstrates the efficacy of the proposed TPTS/EA algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPTS/EA algorithm is able to improve the previous best known results for 2 instances, while matching the previous best-known solutions for 130 instances. We also provide experimental evidences to highlight the beneficial effect of several important components in our TPTS/EA algorithm.     

A hybrid iterated local search heuristic for the maximum weight independent set problem

This paper presents a hybrid iterated local search (ILS) algorithm for the maximum weight independent set (MWIS) problem, a generalization of the classical maximum independent set problem. Two efficient neighborhood structures are proposed and they are explored using the variable neighborhood descent procedure. Moreover, we devise a perturbation mechanism that dynamically adjusts the balance between intensification and diversification during the search. The proposed algorithm was tested on two well-known benchmarks (DIMACS-W and BHOSLIB-W) and the results obtained were compared with those found by state-of-the-art heuristics and exact methods. Our heuristic outperforms the best-known heuristic for the MWIS as well as the best heuristics for the maximum weight clique problem. The results also show that the hybrid ILS was capable of finding all known optimal solutions in milliseconds.     

Local search for the maximum k-plex problem

Journal of Heuristics - The maximum k-plex problem is an important, computationally complex graph based problem. In this study an effective k-plex local search (KLS) is presented for solving this...     

Iterated greedy for the maximum diversity problem

The problem of choosing a subset of elements with maximum diversity from a given set is known as the maximum diversity problem. Many algorithms and methods have been proposed for this hard combinatorial problem, including several highly sophisticated procedures. By contrast, in this paper we present a simple iterated greedy metaheuristic that generates a sequence of solutions by iterating over a greedy construction heuristic using destruction and construction phases. Extensive computational experiments reveal that the proposed algorithm is highly effective as compared to the best-so-far metaheuristics for the problem under consideration.     

New heuristics for the maximum diversity problem

The maximum diversity problem (MDP) consists of identifying, in a population, a subset of elements, characterized by a set of attributes, that present the most diverse characteristics among the elements of the subset. The identification of such solution is an NP-hard problem. Some heuristics are available to obtain approximate solutions for this problem. In this paper, we propose different GRASP heuristics for the MDP, using distinct construction procedures and including a path-relinking technique. Performance comparison among related work and the proposed heuristics is provided. Experimental results show that the new GRASP heuristics are quite robust and are able to find high-quality solutions in reasonable computational times. G.C. Silva’s work sponsored by CAPES MSc scholarship. L.S. Ochi’s work sponsored by CNPq research grants 304103/2003-9 and 550059/2005-9. S.L. Martins’s work sponsored by CNPq research grant 475124/03-0. A. Plastino’s work sponsored by CNPq research grants 300879/00-8 and 475124/03-0.     

Hybrid heuristics for the maximum diversity problem

The maximum diversity problem presents a challenge to solution methods based on heuristic optimization. We undertake the development of hybrid procedures within the scatter search framework with the goal of uncovering the most effective designs to tackle this difficult but important problem. Our research revealed the effectiveness of adding simple memory structures (based on recency and frequency) to key scatter search mechanisms. Our extensive experiments and related statistical tests show that the most effective scatter search variant outperforms state-of-the-art methods.     

Guided construction search metaheuristics for the capacitated p-median problem with single source constraint

In the capacitated p-median problem with single source constraint, also known as the capacitated clustering problem, a given set of n weighted points is to be partitioned into p clusters such that the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centres that minimizes the total scatter of points allocated to these clusters. In this paper, a (λ, μ)-interchange neighbourhood based on the concept of λ-interchange of points restricted to μ-adjacent clusters is proposed. Structural properties of centres are identified and exploited to derive special data structures for their efficient evaluations. Different search and selection strategies including the variable neighbourhood search descent with respect to μ-nearest points are investigated. The most efficient strategies are then embedded in a guided construction search metaheuristic framework based either on a periodic local search procedure or a greedy random adaptive search procedure to solve the problem. Computational experience is reported on a standard set of benchmarks. The computational experience demonstrates the competitive performance of the proposed algorithms when compared to the best-known procedures in the literature in terms of solution quality and computational requirement.     

Multi-neighborhood tabu search for the maximum weight clique problem

Given an undirected graph G=(V,E) with vertex set V={1,??,n} and edge set E?V×V. Let w:V??Z ⁺ be a weighting function that assigns to each vertex i??V a positive integer. The maximum weight clique problem (MWCP) is to determine a clique of maximum weight. This paper introduces a tabu search heuristic whose key features include a combined neighborhood and a dedicated tabu mechanism using a randomized restart strategy for diversification. The proposed algorithm is evaluated on a total of 136 benchmark instances from different sources (DIMACS, BHOSLIB and set packing). Computational results disclose that our new tabu search algorithm outperforms the leading algorithm for the maximum weight clique problem, and in addition rivals the performance of the best methods for the unweighted version of the problem without being specialized to exploit this problem class.     

Survey and unification of local search techniques in metaheuristics for multi-objective combinatorial optimisation

Metaheuristics are algorithms that have proven their efficiency on multi-objective combinatorial optimisation problems. They often use local search techniques, either at their core or as intensification mechanisms, to obtain a well-converged and diversified final result. This paper surveys the use of local search techniques in multi-objective metaheuristics and proposes a general structure to describe and unify their underlying components. This structure can instantiate most of the multi-objective local search techniques and algorithms in literature.     

Iterated local search for the quadratic assignment problem

Iterated local search (ILS) is a simple and powerful stochastic local search method. This article presents and analyzes the application of ILS to the quadratic assignment problem (QAP). We justify the potential usefulness of an ILS approach to this problem by an analysis of the QAP search space. However, an analysis of the run-time behavior of a basic ILS algorithm reveals a stagnation behavior which strongly compromises its performance. To avoid this stagnation behavior, we enhance the ILS algorithm using acceptance criteria that allow moves to worse local optima and we propose population-based ILS extensions. An experimental evaluation of the enhanced ILS algorithms shows their excellent performance when compared to other state-of-the-art algorithms for the QAP.     

On local search for the generalized graph coloring problem

Given an edge-weighted graph and an integer k, the generalized graph coloring problem is the problem of partitioning the vertex set into k subsets so as to minimize the total weight of the edges that are included in a single subset. We recall a result on the equivalence between Karush-Kuhn-Tucker points for a quadratic programming formulation and local optima for the simple flip-neighborhood. We also show that the quality of local optima with respect to a large class of neighborhoods may be arbitrarily bad and that some local optima may be hard to find.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

Diversification strategies in tabu search algorithms for the maximum clique problem

The purpose of this study is to develop some understanding of the benefits that can be derived from the inclusion of diversification strategies in tabu search methods. To do so, we discuss the implementation of various diversification strategies in several tabu search heuristics developed for the maximum clique problem. Computational results on a large set of randomly generated test problems are reported and compared to assess the impact of these techniques on solution quality and running time.