A fast and elitist multiobjective genetic algorithm: NSGA-II

Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN³) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN²) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed     

Comparison of multiobjective evolutionary algorithms: empirical results

In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search.     

A comparison of multiobjective depth-first algorithms

Many real world problems involve several, usually conflicting, objectives. Multiobjective analysis deals with these problems locating trade-offs between different optimal solutions. Regarding graph search problems, several algorithms based on best-first and depth-first approaches have been proposed to return the set of all Pareto optimal solutions. This article presents a detailed comparison between two representatives of multiobjective depth-first algorithms, PIDMOA* and MO-DF-BnB. Both of them extend previous single-objective search algorithms with linear-space requirements to the multiobjective case. Experimental analyses on their time performance over tree-shaped search spaces are presented. The results clarify the fitness of both algorithms to parameters like the number or depth of goal nodes.     

The balance between proximity and diversity in multiobjective evolutionary algorithms

Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multiobjective optimization problems. Especially more recent multiobjective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multiobjective problems. While we point out the most important aspects for designing competent MOEAs in this paper, we also indicate the inherent multiobjective tradeoff in multiobjective optimization between proximity and diversity preservation. We discuss the impact of this tradeoff on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate nondomination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.     

Static and run-time algorithms for all-to-many personalizedcommunication on permutation networks

With the advent of new routing methods, the distance that a message is sent is becoming relatively less and less important. Thus, assuming no link contention, permutation seems to be an efficient collective communication primitive. In this paper, we present several algorithms for decomposing all-to-many personalized communication into a set of disjoint partial permutations. We discuss several algorithms and study their effectiveness from the view of static scheduling as well as run-time scheduling. An approximate analysis shows that with n processors, and assuming that every processor sends and receives d messages to random destinations, our algorithm can perform the scheduling in O(dn In d) time, on average, and can use an expected number of d+log d partial permutations to carry out the communication. We present experimental results of our algorithms on the CM-5     

Probabilistic logic under coherence: complexity and algorithms

In previous work [V. Biazzo, A. Gilio, T. Lukasiewicz and G. Sanfilippo, Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P, Journal of Applied Non-Classical Logics 12(2) (2002) 189–213.], we have explored the relationship between probabilistic reasoning under coherence and model-theoretic probabilistic reasoning. In particular, we have shown that the notions of g-coherence and of g-coherent entailment in probabilistic reasoning under coherence can be expressed by combining notions in model-theoretic probabilistic reasoning with concepts from default reasoning. In this paper, we continue this line of research. Based on the above semantic results, we draw a precise picture of the computational complexity of probabilistic reasoning under coherence. Moreover, we introduce transformations for probabilistic reasoning under coherence, which reduce an instance of deciding g-coherence or of computing tight intervals under g-coherent entailment to a smaller problem instance, and which can be done very efficiently. Furthermore, we present new algorithms for deciding g-coherence and for computing tight intervals under g-coherent entailment, which reformulate previous algorithms using terminology from default reasoning. They are based on reductions to standard problems in model-theoretic probabilistic reasoning, which in turn can be reduced to linear optimization problems. Hence, efficient techniques for model-theoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence (for example, column generation techniques). We describe several such techniques, which transform problem instances in model-theoretic probabilistic reasoning into smaller problem instances. We also describe a technique for obtaining a reduced set of variables for the associated linear optimization problems in the conjunctive case, and give new characterizations of this reduced set as a set of non-decomposable variables, and using the concept of random gain. This paper is a substantially extended and revised version of a preliminary paper that appeared in: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA '01), pp. 51–61, 2001.     

Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.     

Nonmonotonic reasoning: from complexity to algorithms

The purpose of this paper is to outline various results regarding the computational complexity and the algorithms of nonmonotonic entailment in different coherence‐based approaches. Starting from a (non necessarily consistent) belief base E and a pre‐order on E, we first present different mechanisms for selecting preferred consistent subsets. Then we present different entailment principles in order to manage these multiple subsets. The crossing point of each generation mechanism m and each entailment principle p defines an entailment relation which we study from the computational complexity point of view. The results are not very encouraging since the complexity of all these nonmonotonic entailment relations is, in most restricted languages, larger than the complexity of monotonic entailment. So, we decided to extend Binary Decision Diagrams technics, which are well suited to the task of solving NP‐hard logic‐based problems. Both theoretical and experimental results are described along this line in the last sections. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the complexity of admissible search algorithms

This paper analyzes the complexity of heuristic search algorithms, i.e. algorithms which find the shortest path in a graph by using an estimate to guide the search. In particular, algorithm A¹, due to Hart, Nilsson and Raphael, is shown to require O(2^N) steps, in the worst case, for searching a graph with N nodes, if the so called “consistency assumption” does not hold for the estimate. Furthermore, a new search algorithm is presented which runs in O(N²) steps in the worst case and which never requires more steps than A¹.     

State-variable planning under structural restrictions: algorithms and complexity

Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying state-transition graph. The exponential size of this graph, though, makes such restrictions costly to test. Hence, we propose an intermediate approach, using a state-variable model for planning and defining restrictions on the separate state-transition graphs for each state variable. We identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions. The algorithm has been implemented and it outperforms Graphplan on a number of test instances. In addition, we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions. This complexity map considers both the optimal and non-optimal plan generation problem.     

Metaqueries: Semantics, complexity, and efficient algorithms

Metaquery (metapattern) is a data mining tool which is useful for learning rules involving more than one relation in the database. The notion of a metaquery has been proposed as a template or a second-order proposition in a language that describes the type of pattern to be discovered. This tool has already been successfully applied to several real-world applications.In this paper we advance the state of the art in metaquery research in several ways. First, we argue that the notion of a support value for metaqueries, where a support value is intuitively some indication to the relevance of the rules to be discovered, is not adequately defined in the literature, and, hence, propose our own definition. Second, we analyze some of the related computational problems, classify them as NP-hard and point out some tractable cases. Third, we propose some efficient algorithms for computing support and present preliminary experimental results that indicate the usefulness of our algorithms.     

Formation control: The van loan scheme and other algorithms

The subject of this note is the algorithm of evolution of a point set on the plane devised in [5], which drives the whole set to a certain regular configuration. Generalizations of the algorithm are proposed and analyzed, certain new properties are studied, connections to formation control methods is discussed, and new simpler algorithms of this sort are proposed.     

Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The T t-S problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359–387) by showing that, for any t4, T t-S is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t+2 (if t is odd). Then we point out that every chordal graph of diameter at most t−1 (respectively, t−2) admits a tree t-spanner whenever t2 is even (respectively, t3 is odd), and such a tree spanner can be constructed in linear time.

The complexity status of T 3-S still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide T 3-S efficiently.     

Multicommodity flows over time: Efficient algorithms and complexity

Flow variation over time is an important feature in network flow problems arising in various applications such as road or air traffic control, production systems, communication networks (e.g. the Internet) and financial flows. The common characteristic are networks with capacities and transit times on the arcs which specify the amount of time it takes for flow to travel through a particular arc. Moreover, in contrast to static flow problems, flow values on arcs may change with time in these networks.     

Particle computation: complexity,algorithms, and logic

Natural Computing - We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal...     

Generalized kolmogorov complexity and other dual complexity measures

We construct and analyze some dual complexity measures that indicate the time it takes to obtain the desired object. The existence of optimal dual complexity measures is established. Various relations between dual measures and complexities are determined. The relationship of these measures to program quality is demonstrated.Translated from Kibernetika, No. 4, pp. 21–29, 54, July–August, 1990.     

The trade-off between the additive complexity and the asynchronicity of linear and bilinear algorithms

It is shown that every linear algorithm that computes a set of Q linearly independent linear forms in k variables must involve at least Q log₂ k - H additions and subtractions where H is a quantity that measures the asynchronicity of the algorithm. The result is extended to the evaluation of a set of bilinear forms.