An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS$_o$ performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS$_o$ that is suitable for solving the minimum weight dominating set problem. The application of RLS$_o$ in graph mining is also briefly demonstrated

Symmetry Breaking by Metaheuristic Search

Several methods exist for breaking symmetry in constraint problems, but most potentially suffer from high memory requirements, high computational overhead, or both. We describe a new partial symmetry breaking method that can be applied to arbitrary variable/value symmetries. It models dominance detection as a nonstationary optimisation problem, and solves it by resource-bounded metaheuristic search in the symmetry group. It has low memory requirement and computational overhead, yet in preliminary experiments on BIBD design it breaks most symmetries

A Simple Approach to Dynamic Optimisation of Flexible Optical Networks with Practical Application

This paper provides an initial introduction to, and definition of, the ‘Dynamically Powered Relays for a Flexible Optical Network’ (DPR-FON) problem for opto-electro-optical (OEO) regenerators used in optical networks. In such networks, optical transmission parameters can be varied dynamically as traffic patterns change. This will provide different bandwidths, but also change the regeneration limits as a result. To support this flexibility, OEOs (‘relays’) may be switched on and off as required, thus saving power. DPR-FON is shown to be NP-complete; consequently, solving such a dynamic problem in real-time requires a fast heuristic capable of delivering an acceptable approximation to the optimal configuration with low complexity. In this paper, just such an algorithm is developed, implemented, and evaluated against more computationally-demanding alternatives for two known cases. A number of real-world extensions are considered as the paper develops, combining to produce the ‘Generalised Dynamically Powered Relays for a Flexible Optical Network’ (GDPR-FON) problem. This, too, is analysed and an associated fast heuristic proposed, along with an exploration of the further research that is required

ND-Tree-based update: a Fast Algorithm for the Dynamic Non-Dominance Problem

In this paper we propose a new method called ND-Tree-based update (or shortly ND-Tree) for the dynamic non-dominance problem, i.e. the problem of online update of a Pareto archive composed of mutually non-dominated points. It uses a new ND-Tree data structure in which each node represents a subset of points contained in a hyperrectangle defined by its local approximate ideal and nadir points. By building subsets containing points located close in the objective space and using basic properties of the local ideal and nadir points we can efficiently avoid searching many branches in the tree. ND-Tree may be used in multiobjective evolutionary algorithms and other multiobjective metaheuristics to update an archive of potentially non-dominated points. We prove that the proposed algorithm has sub-linear time complexity under mild assumptions. We experimentally compare ND-Tree to the simple list, Quad-tree, and M-Front methods using artificial and realistic benchmarks with up to 10 objectives and show that with this new method substantial reduction of the number of point comparisons and computational time can be obtained. Furthermore, we apply the method to the non-dominated sorting problem showing that it is highly competitive to some recently proposed algorithms dedicated to this problem.Comment: 15 pages, 21 figures, 3 table