Improved Runtime Results for Simple Randomised Search Heuristics on Linear Functions with a Uniform Constraint

In the last decade remarkable progress has been made in development of suitable proof techniques for analysing randomised search heuristics. The theoretical investigation of these algorithms on classes of functions is essential to the understanding of the underlying stochastic process. Linear functions have been traditionally studied in this area resulting in tight bounds on the expected optimisation time of simple randomised search algorithms for this class of problems. Recently, the constrained version of this problem has gained attention and some theoretical results have also been obtained on this class of problems. In this paper we study the class of linear functions under uniform constraint and investigate the expected optimisation time of Randomised Local Search (RLS) and a simple evolutionary algorithm called (1+1) EA. We prove a tight bound of $\Theta(n^2)$ for RLS and improve the previously best known upper bound of (1+1) EA from $O(n^2 \log (Bw_{\max}))$ to $O(n^2\log B)$ in expectation and to $O(n^2 \log n)$ with high probability, where $w_{\max}$ and $B$ are the maximum weight of the linear objective function and the bound of the uniform constraint, respectively. Also, we obtain a tight bound of $O(n^2)$ for the (1+1) EA on a special class of instances. We complement our theoretical studies by experimental investigations that consider different values of $B$ and also higher mutation rates that reflect the fact that $2$-bit flips are crucial for dealing with the uniform constraint.

Runtime Analysis of RLS and EA for the Dynamic Weighted Vertex Cover Problem

In this paper, we perform theoretical analyses on the behaviour of an evolutionary algorithm and a randomised search algorithm for the dynamic vertex cover problem based on its dual formulation. The dynamic vertex cover problem has already been theoretically investigated to some extent and it has been shown that using its dual formulation to represent possible solutions can lead to a better approximation behaviour. We improve some of the existing results, i.e. we find a linear expected re-optimization time for a (1+1) EA to re-discover a 2-approximation when edges are dynamically deleted from the graph. Furthermore, we investigate a different setting for applying the dynamism to the problem, in which a dynamic change happens at each step with a probability $P_D$. We also expand these analyses to the weighted vertex cover problem, in which weights are assigned to vertices and the goal is to find a cover set with minimum total weight. Similar to the classical case, the dynamic changes that we consider on the weighted vertex cover problem are adding and removing edges to and from the graph. We aim at finding a maximal solution for the dual problem, which gives a 2-approximate solution for the vertex cover problem. This is equivalent to the maximal matching problem for the classical vertex cover problem.

Analysis of Baseline Evolutionary Algorithms for the Packing While Travelling Problem

Although the performance of base-line Evolutionary Algorithms (EAs) on linear functions has been studied rigorously, the same theoretical analyses on non-linear objectives are still far behind. In this paper, variations of the Packing While Travelling (PWT), also known as a non-linear knapsack problem, is considered to address this gap. We investigate PWT for two cities with correlated weights and profits using single-objective and multi-objective algorithms. Our results show that RLS finds the optimal solution in $O(n^3)$ expected time while the GSEMO enhanced with a specific selection operator to deal with exponential population size, calculates all the Pareto front solutions in the same expected time. In the case of uniform weights, (1+1)~EA is able to find the optimal solution in expected time $O(n^2\log{(\max\{n,p_{max}\})})$, where $p_{max}$ is the largest profit of the given items. We also validate the theoretical results using practical experiments and present estimation for expected running time according to the experiments.

Analysis of Evolutionary Algorithms in Dynamic and Stochastic Environments

Many real-world optimization problems occur in environments that change dynamically or involve stochastic components. Evolutionary algorithms and other bio-inspired algorithms have been widely applied to dynamic and stochastic problems. This survey gives an overview of major theoretical developments in the area of runtime analysis for these problems. We review recent theoretical studies of evolutionary algorithms and ant colony optimization for problems where the objective functions or the constraints change over time. Furthermore, we consider stochastic problems under various noise models and point out some directions for future research.

Parameterized Analysis of Multi-objective Evolutionary Algorithms and the Weighted Vertex Cover Problem

A rigorous runtime analysis of evolutionary multi-objective optimization for the classical vertex cover problem in the context of parameterized complexity analysis has been presented by Kratsch and Neumann (2013). In this paper, we extend the analysis to the weighted vertex cover problem and provide a fixed parameter evolutionary algorithm with respect to OPT, the cost of the the optimal solution for the problem. Moreover, using a diversity mechanisms, we present a multi-objective evolutionary algorithm that finds a 2-approximation in expected polynomial time and introduce a population-based evolutionary algorithm which finds a $(1+\varepsilon)$-approximation in expected time $O(n\cdot 2^{\min \{n,2(1- \varepsilon)OPT \}} + n^3)$.

A Parameterized Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms

Bi-level optimisation problems have gained increasing interest in the field of combinatorial optimisation in recent years. With this paper, we start the runtime analysis of evolutionary algorithms for bi-level optimisation problems. We examine two NP-hard problems, the generalised minimum spanning tree problem (GMST), and the generalised travelling salesman problem (GTSP) in the context of parameterised complexity. For the generalised minimum spanning tree problem, we analyse the two approaches presented by Hu and Raidl (2012) with respect to the number of clusters that distinguish each other by the chosen representation of possible solutions. Our results show that a (1+1) EA working with the spanning nodes representation is not a fixed-parameter evolutionary algorithm for the problem, whereas the global structure representation enables to solve the problem in fixed-parameter time. We present hard instances for each approach and show that the two approaches are highly complementary by proving that they solve each other's hard instances very efficiently. For the generalised travelling salesman problem, we analyse the problem with respect to the number of clusters in the problem instance. Our results show that a (1+1) EA working with the global structure representation is a fixed-parameter evolutionary algorithm for the problem.