Drift Analysis with Fitness Levels for Elitist Evolutionary Algorithms

The fitness level method is a popular tool for analyzing the computation time of elitist evolutionary algorithms. Its idea is to divide the search space into multiple fitness levels and estimate lower and upper bounds on the computation time using transition probabilities between fitness levels. However, the lower bound generated from this method is often not tight. To improve the lower bound, this paper rigorously studies an open question about the fitness level method: what are the tightest lower and upper time bounds that can be constructed based on fitness levels? To answer this question, drift analysis with fitness levels is developed, and the tightest bound problem is formulated as a constrained multi-objective optimization problem subject to fitness level constraints. The tightest metric bounds from fitness levels are constructed and proven for the first time. Then the metric bounds are converted into linear bounds, where existing linear bounds are special cases. This paper establishes a general framework that can cover various linear bounds from trivial to best coefficients. It is generic and promising, as it can be used not only to draw the same bounds as existing ones, but also to draw tighter bounds, especially on fitness landscapes where shortcuts exist. This is demonstrated in the case study of the (1+1) EA maximizing the TwoPath function

A comparative runtime analysis of heuristic algorithms for satisfiability problems

AbstractThe satisfiability problem is a basic core NP-complete problem. In recent years, a lot of heuristic algorithms have been developed to solve this problem, and many experiments have evaluated and compared the performance of different heuristic algorithms. However, rigorous theoretical analysis and comparison are rare. This paper analyzes and compares the expected runtime of three basic heuristic algorithms: RandomWalk, (1+1) EA, and hybrid algorithm. The runtime analysis of these heuristic algorithms on two 2-SAT instances shows that the expected runtime of these heuristic algorithms can be exponential time or polynomial time. Furthermore, these heuristic algorithms have their own advantages and disadvantages in solving different SAT instances. It also demonstrates that the expected runtime upper bound of RandomWalk on arbitrary k-SAT (k⩾3) is O((k−1)n), and presents a k-SAT instance that has Θ((k−1)n) expected runtime bound

A Comparison of GAs Penalizing Infeasible Solutions and Repairing Infeasible Solutions on the 0-1 Knapsack Problem

Constraints exist in almost every optimization problem. Different constraint handling techniques have been incorporated with genetic algorithms (GAs), however most of current studies are based on computer experiments. An example is Michalewicz\u27s comparison among GAs using different constraint handling techniques on the 0-1 knapsack problem. The following phenomena are observed in experiments: 1) the penalty method needs more generations to find a feasible solution to the restrictive capacity knapsack than the repair method; 2) the penalty method can find better solutions to the average capacity knapsack. Such observations need a theoretical explanation. This paper aims at providing a theoretical analysis of Michalewicz\u27s experiments. The main result of the paper is that GAs using the repair method are more efficient than GAs using the penalty method on both restrictive capacity and average capacity knapsack problems. This result of the average capacity is a little different from Michalewicz\u27s experimental results. So a supplemental experiment is implemented to support the theoretical claim. The results confirm the general principle pointed out by Coello: a better constraint-handling approach should tend to exploit specific domain knowledge

DsMtGCN: A Direction-sensitive Multi-task framework for Knowledge Graph Completion

To solve the inherent incompleteness of knowledge graphs (KGs), numbers of knowledge graph completion (KGC) models have been proposed to predict missing links from known triples. Among those, several works have achieved more advanced results via exploiting the structure information on KGs with Graph Convolutional Networks (GCN). However, we observe that entity embeddings aggregated from neighbors in different directions are just simply averaged to complete single-tasks by existing GCN based models, ignoring the specific requirements of forward and backward sub-tasks. In this paper, we propose a Direction-sensitive Multi-task GCN (DsMtGCN) to make full use of the direction information, the multi-head self-attention is applied to specifically combine embeddings in different directions based on various entities and sub-tasks, the geometric constraints are imposed to adjust the distribution of embeddings, and the traditional binary cross-entropy loss is modified to reflect the triple uncertainty. Moreover, the competitive experiments results on several benchmark datasets verify the effectiveness of our model

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