Solving the minimum labelling spanning tree problem using hybrid local search

Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics (Modified Genetic Algorithm (MGA) and Pilot Method (PILOT)) have been proposed and analyzed. A hybrid local search method, that we call Group-Swap Variable Neighbourhood Search (GS-VNS), is proposed in this paper. It is obtained by combining two classic metaheuristics: Variable Neighbourhood Search (VNS) and Simulated Annealing (SA). Computational experiments show that GS-VNS outperforms MGA and PILOT. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristic

Variable neighbourhood search for the minimum labelling Steiner tree problem

We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to represent many real world problems in telecommunications and multimodal transportation networks. Several metaheuristics are proposed and evaluated. The approaches are compared to the widely adopted Pilot Method and it is shown that the Variable Neighbourhood Search metaheuristic is the most effective approach to the problem, obtaining high quality solutions in short computational running times

The development and application of metaheuristics for problems in graph theory: A computational study

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.It is known that graph theoretic models have extensive application to real-life discrete optimization problems. Many of these models are NP-hard and, as a result, exact methods may be impractical for large scale problem instances. Consequently, there is a great interest in developing e±cient approximate methods that yield near-optimal solutions in acceptable computational times. A class of such methods, known as metaheuristics, have been proposed with success. This thesis considers some recently proposed NP-hard combinatorial optimization problems formulated on graphs. In particular, the min- imum labelling spanning tree problem, the minimum labelling Steiner tree problem, and the minimum quartet tree cost problem, are inves- tigated. Several metaheuristics are proposed for each problem, from classical approximation algorithms to novel approaches. A compre- hensive computational investigation in which the proposed methods are compared with other algorithms recommended in the literature is reported. The results show that the proposed metaheuristics outper- form the algorithms recommended in the literature, obtaining optimal or near-optimal solutions in short computational running times. In addition, a thorough analysis of the implementation of these methods provide insights for the implementation of metaheuristic strategies for other graph theoretic problems

Batch Informed Trees (BIT*): Informed Asymptotically Optimal Anytime Search

Path planning in robotics often requires finding high-quality solutions to continuously valued and/or high-dimensional problems. These problems are challenging and most planning algorithms instead solve simplified approximations. Popular approximations include graphs and random samples, as respectively used by informed graph-based searches and anytime sampling-based planners. Informed graph-based searches, such as A*, traditionally use heuristics to search a priori graphs in order of potential solution quality. This makes their search efficient but leaves their performance dependent on the chosen approximation. If its resolution is too low then they may not find a (suitable) solution but if it is too high then they may take a prohibitively long time to do so. Anytime sampling-based planners, such as RRT*, traditionally use random sampling to approximate the problem domain incrementally. This allows them to increase resolution until a suitable solution is found but makes their search dependent on the order of approximation. Arbitrary sequences of random samples approximate the problem domain in every direction simultaneously and but may be prohibitively inefficient at containing a solution. This paper unifies and extends these two approaches to develop Batch Informed Trees (BIT*), an informed, anytime sampling-based planner. BIT* solves continuous path planning problems efficiently by using sampling and heuristics to alternately approximate and search the problem domain. Its search is ordered by potential solution quality, as in A*, and its approximation improves indefinitely with additional computational time, as in RRT*. It is shown analytically to be almost-surely asymptotically optimal and experimentally to outperform existing sampling-based planners, especially on high-dimensional planning problems.Comment: International Journal of Robotics Research (IJRR). 32 Pages. 16 Figure

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

Near-Shortest Path Routing in Hybrid Communication Networks

Hybrid networks, i.e., networks that leverage different means of communication, become ever more widespread. To allow theoretical study of such networks, [Augustine et al., SODA\u2720] introduced the HYBRID model, which is based on the concept of synchronous message passing and uses two fundamentally different principles of communication: a local mode, which allows every node to exchange one message per round with each neighbor in a local communication graph; and a global mode where any pair of nodes can exchange messages, but only few such exchanges can take place per round. A sizable portion of the previous research for the HYBRID model revolves around basic communication primitives and computing distances or shortest paths in networks. In this paper, we extend this study to a related fundamental problem of computing compact routing schemes for near-shortest paths in the local communication graph. We demonstrate that, for the case where the local communication graph is a unit-disc graph with n nodes that is realized in the plane and has no radio holes, we can deterministically compute a routing scheme that has constant stretch and uses labels and local routing tables of size O(log n) bits in only O(log n) rounds

Performance Analysis of Evolutionary Algorithms for the Minimum Label Spanning Tree Problem

Some experimental investigations have shown that evolutionary algorithms (EAs) are efficient for the minimum label spanning tree (MLST) problem. However, we know little about that in theory. As one step towards this issue, we theoretically analyze the performances of the (1+1) EA, a simple version of EAs, and a multi-objective evolutionary algorithm called GSEMO on the MLST problem. We reveal that for the MLST$_{b}$ problem the (1+1) EA and GSEMO achieve a $\frac{b+1}{2}$-approximation ratio in expected polynomial times of $n$ the number of nodes and $k$ the number of labels. We also show that GSEMO achieves a $(2ln(n))$-approximation ratio for the MLST problem in expected polynomial time of $n$ and $k$. At the same time, we show that the (1+1) EA and GSEMO outperform local search algorithms on three instances of the MLST problem. We also construct an instance on which GSEMO outperforms the (1+1) EA

Particle swarm optimization for the Steiner tree in graph and delay-constrained multicast routing problems

This paper presents the first investigation on applying a particle swarm optimization (PSO) algorithm to both the Steiner tree problem and the delay constrained multicast routing problem. Steiner tree problems, being the underlining models of many applications, have received significant research attention within the meta-heuristics community. The literature on the application of meta-heuristics to multicast routing problems is less extensive but includes several promising approaches. Many interesting research issues still remain to be investigated, for example, the inclusion of different constraints, such as delay bounds, when finding multicast trees with minimum cost. In this paper, we develop a novel PSO algorithm based on the jumping PSO (JPSO) algorithm recently developed by Moreno-Perez et al. (Proc. of the 7th Metaheuristics International Conference, 2007), and also propose two novel local search heuristics within our JPSO framework. A path replacement operator has been used in particle moves to improve the positions of the particle with regard to the structure of the tree. We test the performance of our JPSO algorithm, and the effect of the integrated local search heuristics by an extensive set of experiments on multicast routing benchmark problems and Steiner tree problems from the OR library. The experimental results show the superior performance of the proposed JPSO algorithm over a number of other state-of-the-art approaches

Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques

We present exact mixed integer programming approaches including branch-and-cut and branch-and-cut-and-price for the minimum label spanning tree problem as well as a variant of it having multiple labels assigned to each edge. We compare formulations based on network flows and directed connectivity cuts. Further, we show how to use odd-hole inequalities and additional inequalities to strengthen the formulation. Label variables can be added dynamically to the model in the pricing step. Primal heuristics are incorporated into the framework to speed up the overall solution process. After a polyhedral comparison of the involved formulations, comprehensive computational experiments are presented in order to compare and evaluate the underlying formulations and the particular algorithmic building blocks of the overall branch-and-cut- (and-price) framework