Given a set of @math elements separated by a pairwise distance matrix, the minimum differential dispersion problem (Min-Diff DP) aims to identify a subset of m elements (m < n) such that the difference between the maximum sum and the minimum sum of the inter-element distances between any two chosen elements is minimized. We propose an effective iterated local search (denoted by ILS_MinDiff) for Min-Diff DP. To ensure an effective exploration and exploitation of the search space, the proposed ILS_MinDiff algorithm iterates through three sequential search phases: a fast descent-based neighborhood search phase to find a local optimum from a given starting solution, a local optima exploring phase to visit nearby high-quality solutions around a given local optimum, and a local optima escaping phase to move away from the current search region. Experimental results on six data sets of 190 benchmark instances demonstrate that ILS_MinDiff competes favorably with the state-of-the-art algorithms by finding 130 improved best results (new upper bounds).
Based on the general ILS framework, several variants and extended approaches have recently been proposed in the literature, of which two representative examples are breakout local search (BLS) @cite_35 @cite_5 and three-phase search (TPS) @cite_18 . The effectiveness of BLS and TPS have been verified on a variety of hard optimization problems and applications (see examples of Table ). In the following, we present a brief review of these ILS variants.
@cite_35 @cite_5 @cite_18
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In this paper, we propose the first heuristic approach for the vertex separator problem (VSP), based on Breakout Local Search (BLS). BLS is a recent meta-heuristic that follows the general framework of the popular Iterated Local Search (ILS) with a particular focus on the perturbation strategy. Based on some relevant information on search history, it tries to introduce the most suitable degree of diversification by determining adaptively the number and type of moves for the next perturbation phase. The proposed heuristic is highly competitive with the exact state-of-art approaches from the literature on the current VSP benchmark. Moreover, we present for the first time computational results for a set of large graphs with up to 3000 vertices, which constitutes a new challenging benchmark for VSP approaches. Given an undirected graph with costs associated with each edge as well as each pair of edges, the quadratic minimum spanning tree problem (QMSTP) consists of determining a spanning tree of minimum cost. QMSTP is useful to model many real-life network design applications. We propose a three-phase search approach named TPS for solving QMSTP, which organizes the search process into three distinctive phases which are iterated: (1) a descent neighborhood search phase using two move operators to reach a local optimum from a given starting solution, (2) a local optima exploring phase to discover nearby local optima within a given regional area, and (3) a perturbation-based diversification phase to jump out of the current regional search area. TPS also introduces a pre-estimation criterion to significantly improve the efficiency of neighborhood evaluation, and develops a new swap-vertex neighborhood (as well as a swap-vertex based perturbation operator) which prove to be quite powerful for solving a series of special instances with particular structures. Computational experiments based on 7 sets of 659 popular benchmarks show that TPS produces highly competitive results compared to the best performing approaches in the literature. TPS discovers improved best known results (new upper bounds) for 33 open instances and matches the best known results for all the remaining instances. Critical elements and parameters of the TPS algorithm are analyzed to understand its behavior. HighlightsQMSTP is a general model able to formulate a number of network design problems.We propose a three phase search heuristic (TPS) for this problem.TPS is assessed on 7 groups of 659 representative benchmarks of the literature.TPS finds improved best solutions for 33 challenging instances.TPS finds all the optimal solutions for the 29 instances transformed from the QAP.