Leon Schiller

Bio: Leon Schiller is an academic researcher from University of Potsdam. The author has contributed to research in topics: Computer science & Crossover. The author has co-authored 5 publications. Previous affiliations of Leon Schiller include Hasso Plattner Institute.

Crossover for cardinality constrained optimization

Abstract: In order to understand better how and why crossover can benefit optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1s allowed in the bit string (cardinality constraint). We consider the natural translation of the OneMax test function, a linear function where B bits have a weight of 1 + ε and the remaining bits have a weight of 1. The literature gives a bound of Θ(n2) for the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping both a 1 and a 0. The experimental literature proposes balanced operators, preserving the number of 1s, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n) if n - B = O(1). However, if n - B = Θ(n), we show abound of Ω(n2), just as classic bit flip mutation. Crossover and a simple island model gives O(n2/log n) (uniform crossover) and [EQUATION] (3-ary majority vote crossover). For balanced uniform crossover with Hamming distance maximization for diversity we show a bound of O(n log n). As an additional contribution we analyze and discuss different balanced crossover operators from the literature.

Evolutionary minimization of traffic congestion

Abstract: Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the Multiple-Routes problem, which is given a start and destination and aims at finding up to n different routes that the drivers strategically disperse over, minimizing the overall travel time of the system. Due to the NP-hard nature of the problem, we introduce the Multiple-Routes evolutionary algorithm (MREA) as a heuristic solver. We study several mutation and crossover operators and evaluate them on real-world data of Berlin, Germany. We find that a combination of all operators yields the best result, improving the overall travel time by a factor between 1.8 and 3, in the median, compared to all drivers taking the fastest route. For the base case n = 2, we compare our MREA to the highly tailored optimal solver by Blasius et al. [ATMOS 2020] and show that, in the median, our approach finds solutions of quality at least 99.69% of an optimal solution while only requiring 40 % of the time.

Crossover for Cardinality Constrained Optimization

Abstract: To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length-n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ (n2) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n) if n-B = O(1). However, if n-B = Θ (n), we show a bound of Ω (n2), just as for classic bit mutation. Crossover together with a simple island model gives running times of O(n2 / log n) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O(n log n). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Abstract: A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

Learning Languages with Decidable Hypotheses

Abstract: In language learning in the limit, the most common type of hypothesis is to give an enumerator for a language. This so-called $W$-index allows for naming arbitrary computably enumerable languages, with the drawback that even the membership problem is undecidable. In this paper we use a different system which allows for naming arbitrary decidable languages, namely programs for characteristic functions (called $C$-indices). These indices have the drawback that it is now not decidable whether a given hypothesis is even a legal $C$-index. In this first analysis of learning with $C$-indices, we give a structured account of the learning power of various restrictions employing $C$-indices, also when compared with $W$-indices. We establish a hierarchy of learning power depending on whether $C$-indices are required (a) on all outputs; (b) only on outputs relevant for the class to be learned and (c) only in the limit as final, correct hypotheses. Furthermore, all these settings are weaker than learning with $W$-indices (even when restricted to classes of computable languages). We analyze all these questions also in relation to the mode of data presentation. Finally, we also ask about the relation of semantic versus syntactic convergence and derive the map of pairwise relations for these two kinds of convergence coupled with various forms of data presentation.

Quantum annealing for industry applications: introduction and review

Abstract: Abstract Quantum annealing (QA) is a heuristic quantum optimization algorithm that can be used to solve combinatorial optimization problems. In recent years, advances in quantum technologies have enabled the development of small- and intermediate-scale quantum processors that implement the QA algorithm for programmable use. Specifically, QA processors produced by D-Wave systems have been studied and tested extensively in both research and industrial settings across different disciplines. In this paper we provide a literature review of the theoretical motivations for QA as a heuristic quantum optimization algorithm, the software and hardware that is required to use such quantum processors, and the state-of-the-art applications and proofs-of-concepts that have been demonstrated using them. The goal of our review is to provide a centralized and condensed source regarding applications of QA technology. We identify the advantages, limitations, and potential of QA for both researchers and practitioners from various fields.

What's Wrong with Deep Learning in Tree Search for Combinatorial Optimization

Abstract: Combinatorial optimization lies at the core of many real-world problems. Especially since the rise of graph neural networks (GNNs), the deep learning community has been developing solvers that derive solutions to NP-hard problems by learning the problem-specific solution structure. However, reproducing the results of these publications proves to be difficult. We make three contributions. First, we present an open-source benchmark suite for the NP-hard Maximum Independent Set problem, in both its weighted and unweighted variants. The suite offers a unified interface to various state-of-the-art traditional and machine learning-based solvers. Second, using our benchmark suite, we conduct an in-depth analysis of the popular guided tree search algorithm by Li et al. [NeurIPS 2018], testing various configurations on small and large synthetic and real-world graphs. By re-implementing their algorithm with a focus on code quality and extensibility, we show that the graph convolution network used in the tree search does not learn a meaningful representation of the solution structure, and can in fact be replaced by random values. Instead, the tree search relies on algorithmic techniques like graph kernelization to find good solutions. Thus, the results from the original publication are not reproducible. Third, we extend the analysis to compare the tree search implementations to other solvers, showing that the classical algorithmic solvers often are faster, while providing solutions of similar quality. Additionally, we analyze a recent solver based on reinforcement learning and observe that for this solver, the GNN is responsible for the competitive solution quality.

Crossover for cardinality constrained optimization

Abstract: In order to understand better how and why crossover can benefit optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1s allowed in the bit string (cardinality constraint). We consider the natural translation of the OneMax test function, a linear function where B bits have a weight of 1 + ε and the remaining bits have a weight of 1. The literature gives a bound of Θ(n2) for the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping both a 1 and a 0. The experimental literature proposes balanced operators, preserving the number of 1s, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n) if n - B = O(1). However, if n - B = Θ(n), we show abound of Ω(n2), just as classic bit flip mutation. Crossover and a simple island model gives O(n2/log n) (uniform crossover) and [EQUATION] (3-ary majority vote crossover). For balanced uniform crossover with Hamming distance maximization for diversity we show a bound of O(n log n). As an additional contribution we analyze and discuss different balanced crossover operators from the literature.

Crossover for Cardinality Constrained Optimization

Abstract: To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length-n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ (n2) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n) if n-B = O(1). However, if n-B = Θ (n), we show a bound of Ω (n2), just as for classic bit mutation. Crossover together with a simple island model gives running times of O(n2 / log n) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O(n log n). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.

Can Evolutionary Clustering Have Theoretical Guarantees?

Abstract: —Clustering is a fundamental problem in many areas, which aims to partition a given data set into groups based on some distance measure, such that the data points in the same group are similar while that in different groups are dissimilar. Due to its importance and NP-hardness, a lot of methods have been proposed, among which evolutionary algorithms are a class of popular ones. Evolutionary clustering has found many successful applications, but all the results are empirical, lacking theoretical support. This paper fills this gap by proving that the approximation performance of the GSEMO (a simple multi- objective evolutionary algorithm) for solving the three popular formulations of clustering, i.e., k -center, k -median and k -means, can be theoretically guaranteed. Furthermore, we prove that evolutionary clustering can have theoretical guarantees even when considering fairness, which tries to avoid algorithmic bias, and has recently been an important research topic in machine learning.