Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+\eps) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless $P = NP$) nor to approximate it (unless $NP = BPP$). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0 it calculates a solution with contribution at most (1+\eps) times the minimal contribution with probability at least $(1-\delta)$. Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc

Bringing Order to Special Cases of Klee's Measure Problem

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].Comment: 17 page

Hypervolume based metaheuristics for multiobjective optimization

The purpose of multiobjective optimization is to find solutions that are optimal regarding several goals. In the branch of vector or Pareto optimization all these goals are considered to be of equal importance, so that compromise solutions that cannot be improved regarding one goal without deteriorating in another are Paretooptimal. A variety of quality measures exist to evaluate approximations of the Paretooptimal set generated by optimizers, wherein the hypervolume is the most significant one, making the hypervolume calculation a core problem of multiobjective optimization. This thesis tackles that challenge by providing a new hypervolume algorithm from computational geometry and analyzing the problem’s computational complexity. Evolutionary multiobjective optimization algorithms (EMOA) are state-of-the-art methods for Pareto optimization, wherein the hypervolume-based algorithms belong to the most powerful ones, among them the popular SMS-EMOA. After its promising capabilities have already been demonstrated in first studies, this thesis is dedicated to deeper understand the underlying optimization process of the SMS-EMOA and similar algorithms, in order to specify their performance characteristics. Theoretical analyses are accomplished as far as possible with established and newly developed tools. Beyond the limitations of rigorous scrutiny, insights are gained via thorough experimental investigation. All considered problems are continuous, whereas the algorithms are as well applicable to discrete problems. More precisely, the following topics are concerned. The process of approaching the Pareto-optimal set of points is characterized by the convergence speed, which is analyzed for a general framework of EA with hypervolume selection on several classes of bi-objective problems. The results are achieved by a newly developed concept of linking single and multiobjective optimization. The optimization on the Pareto front, that is turning the population into a set with maximal hypervolume, is considered separately, focusing on the question under which circumstances the steady-state selection of exchanging only one population member suffices to reach a global optimum. We answer this question for different bi-objective problem classes. In a benchmarking on so-called many-objective problems of more than three objectives, the qualification of the SMS-EMOA is demonstrated in comparison to other EMOA, while also studying their cause of failure. Within the mentioned examinations, the choice of the hypervolume’s reference point receives special consideration by exposing its influence. Beyond the study of the SMS-EMOA with default setup, it is analyzed to what extent the performance can be improved by parameter tuning of the EMOA anent to certain problems, focusing on the influence of variation operators. Lastly, an optimization algorithm based on the gradient of the hypervolume is developed and hybridized with the SMS-EMOA

Rail accessibility in Germany: Changing regional disparities between 1990 and 2020

Transport accessibility is an important location factor for households and firms. In the last few decades, technological and social developments have contributed to a reinvigorated role of passenger transport. However, rail accessibility is unevenly distributed in space. The introduction of high-speed rail has furthermore promoted a polarisation of accessibility between metropolises and peripheral areas in some European countries. In this article we analyse the development of rail accessibility at the regional level in Germany between 1990 and 2020 for 266 functional city-regions. Our results show two different facets: The number of regions that are directly connected to one another has decreased, but at the same time the spatial disparities of accessibility have decreased, albeit to a small extent. This development was strongest in East Germany after German reunification and thus largely a consequence of the renovation of the conventional rail infrastructure, not high-speed rail. Nevertheless, it can be concluded that the introduction of high-speed traffic in Germany did not lead to an increase in accessibility disparities. Instead, the accessibility effects of high-speed rail in Germany seem to break the traditional dichotomy between core and periphery.Verkehrliche Erreichbarkeit stellt einen wichtigen Standortfaktor für Haushalte und Unternehmen dar. In den letzten Jahrzehnten haben technologische und soziale Entwicklungen zu einer neuen Attraktivität des Schienenpersonenverkehrs beigetragen. Die Erreichbarkeit über den Schienenverkehr fällt jedoch räumlich sehr unterschiedlich aus. Die Einführung des Hochgeschwindigkeitsverkehrs hat zudem in einigen europäischen Ländern eine Polarisierung der Erreichbarkeit zwischen Metropolen und peripheren Räumen befördert. In diesem Beitrag analysieren wir die Entwicklung der Bahnerreichbarkeit auf regionaler Ebene in Deutschland zwischen 1990 und 2020 für 266 funktionale Stadtregionen. Unsere Ergebnisse zeigen zwei unterschiedliche Facetten: Die Zahl der direkt miteinander verbundenen Regionen hat sich verringert, aber zugleich zeigt sich für die Erreichbarkeit der Bevölkerung eine Abschwächung der räumlichen Disparitäten, wenn auch in geringem Maße. Diese Entwicklung war in Ostdeutschland nach der deutschen Wiedervereinigung am stärksten und damit weitgehend eine Folge der Sanierung der konventionellen Schieneninfrastruktur, nicht des Hochgeschwindigkeitsverkehrs. Dennoch kann der Schluss gezogen werden, dass seine Einführung in Deutschland nicht zur Erhöhung von Erreichbarkeitsdisparitäten geführt hat. Stattdessen scheinen die Erreichbarkeitswirkungen des Hochgeschwindigkeitsverkehrs in Deutschland die traditionelle Dichotomie zwischen Kern und Peripherie zu durchbrechen

Preference Articulation by Means of the R2 Indicator

International audienceIn multi-objective optimization, set-based performance indicators have become the state of the art for assessing the quality of Pareto front approximations. As a consequence, they are also more and more used within the design of multi-objective optimization algorithms. The R2 and the Hypervolume (HV) indicator represent two popular examples. In order to understand the behavior and the approximations preferred by these indicators and algorithms, a comprehensive knowledge of the indicator's properties is required. Whereas this knowledge is available for the HV, we presented a first approach in this direction for the R2 indicator just recently. In this paper, we build upon this knowledge and enhance the considerations with respect to the integration of preferences into the R2 indicator. More specifically, we analyze the effect of the reference point, the domain of the weights, and the distribution of weight vectors on the optimization of μ solutions with respect to the R2 indicator. By means of theoretical findings and empirical evidence, we show the potentials of these three possibilities using the optimal distribution of μ solutions for exemplary setups

Intelligent group movement and selection in realtime strategy games

Movement of groups in realtime strategy games is often a nuisance: Units travel and battle separately, resulting in huge losses and the AI looking dumb. This applies to computer as well as human commanded factions. We suggest to tackle that by using flocking improved by influence-map based pathfinding which leads to a much more natural and intelligent looking behavior. A similar problem occurs if the computer AI has to select groups to combat a specific target: Assignment of units to groups, especially for multiple enemy groups, is often suboptimal when units have very different attack skills. This can be cured by using offline prepared self-organizing feature maps that use all available information for looking up good matches. We demonstrate that these two approaches work well separately, but also that they go together very naturally, thereby leading to an improved and - via parametrization - very flexible group behavior. Opponent AI may be strenghtened that way as well as player-supportive AI. A thorough experimental analysis supports our claims

Multiobjective exploration of the StarCraft map space

This paper presents a search-based method for generating maps for the popular real-time strategy (RTS) game StarCraft. We devise a representation of StarCraft maps suitable for evolutionary search, along with a set of fitness functions based on predicted entertainment value of those maps, as derived from theories of player experience. A multiobjective evolutionary algorithm is then used to evolve complete Star- Craft maps based on the representation and selected fitness functions. The output of this algorithm is a Pareto front approximation visualizing the tradeoff between the several fitness functions used, and where each point on the front represents a viable map. We argue that this method is useful for both automatic and machine-assisted map generation, and in particular that the Pareto fronts are excellent design support tools for human map designers.This research was supported in part by the Danish Research Agency, Ministry of Science, Technology and Innovation; project name: Adaptive Game Content Creation using Computational Intelligence (AGameComIn); project number: 274-09-0083.peer-reviewe

Force-based Cooperative Search Directions in Evolutionary Multi-objective Optimization

International audienceIn order to approximate the set of Pareto optimal solutions, several evolutionary multi-objective optimization (EMO) algorithms transfer the multi-objective problem into several independent single-objective ones by means of scalarizing functions. The choice of the scalarizing functions' underlying search directions, however, is typically problem-dependent and therefore difficult if no information about the problem characteristics are known before the search process. The goal of this paper is to present new ideas of how these search directions can be computed \emph{adaptively} during the search process in a \emph{cooperative} manner. Based on the idea of Newton's law of universal gravitation, solutions attract and repel each other \emph{in the objective space}. Several force-based EMO algorithms are proposed and compared experimentally on general bi-objective $\rho$MNK landscapes with different objective correlations. It turns out that the new approach is easy to implement, fast, and competitive with respect to a $(\mu+\lambda)$-SMS-EMOA variant, in particular if the objectives show strong positive or negative correlations