Tobias Friedrich

Davide Bilò, Keerti Choudhary, Sarel Cohen et al.

Given two vertex sets $S$ and $T$ in a graph, the $ST$-diameter is the maximum $s$-$t$-distance between vertices $s \in S$ and $t \in T$. We study the problem of estimating the $ST$-diameter of graphs that are subject to a small number of transient edge failures. An $f$-edge fault-tolerant $ST$-diameter oracle ($f$-FDO-$ST$) is a data structure that preprocesses a graph $G$, sets $S$, $T$, and a positive integer $f$. When queried with a set $F$ of at most $f$ failing edges, the oracle returns an estimate $\widehat{D}$ of the $ST$-diameter in $G-F$. The oracle is said to have stretch $σ\geq 1$ if $\operatorname{diam}(G{-}F,S,T) \leq \widehat{D} \leq σ\cdot \operatorname{diam}(G{-}F,S,T)$. We design new $f$-FDO-$ST$s by reducing their construction to that of all-pairs and single-source distance sensitivity oracles ($f$-DSOs). These are data structures that estimate the pairwise graph distances, or respectively the distances from a distinguished source, under up to $f$ failures. We obtain several new trade-offs between the size of the $ST$-diameter oracles, their stretch guarantees, query and preprocessing times by combining our black-box reductions with $f$-DSO results from the literature. We further provide a lower bound on the space requirement of approximate $ST$-diameter oracles. We prove that there exists a family of graphs for which any $f$-FDO-$ST$ with sensitivity $f \ge 2$ and stretch better than $5/3$ requires $Ω(n^{3/2})$ bits of space, regardless of the query time.

Tobias Friedrich, Pascal Lenzner, Louise Molitor et al.

Schelling games model the wide-spread phenomenon of residential segregation in metropolitan areas from a game-theoretic point of view. In these games agents of different types each strategically select a node on a given graph that models the residential area to maximize their individual utility. The latter solely depends on the types of the agents on neighboring nodes and it has been a standard assumption to consider utility functions that are monotone in the number of same-type neighbors. This simplifying assumption has recently been challenged since sociological poll results suggest that real-world agents actually favor diverse neighborhoods. We contribute to the recent endeavor of investigating residential segregation models with realistic agent behavior by studying Jump Schelling Games with agents having a single-peaked utility function. In such games, there are empty nodes in the graph and agents can strategically jump to such nodes to improve their utility. We investigate the existence of equilibria and show that they exist under specific conditions. Contrasting this, we prove that even on simple topologies like paths or rings such stable states are not guaranteed to exist. Regarding the game dynamics, we show that improving response cycles exist independently of the position of the peak in the utility function. Moreover, we show high almost tight bounds on the Price of Anarchy and the Price of Stability with respect to the recently proposed degree of integration, which counts the number of agents with a diverse neighborhood and which serves as a proxy for measuring the segregation strength. Last but not least, we show that computing a beneficial state with high integration is NP-complete and, as a novel conceptual contribution, we also show that it is NP-hard to decide if an equilibrium state can be found via improving response dynamics starting from a given initial state.

Katrin Casel, Tobias Friedrich, Martin Schirneck et al.

The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. We discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view. While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable. The most surprising insight to us is the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition.

Davin Jeong, Allison Gunby-Mann, Sarel Cohen et al.

One of the most fundamental graph problems is finding a shortest path from a source to a target node. While in its basic forms the problem has been studied extensively and efficient algorithms are known, it becomes significantly harder as soon as parts of the graph are susceptible to failure. Although one can recompute a shortest replacement path after every outage, this is rather inefficient both in time and/or storage. One way to overcome this problem is to shift computational burden from the queries into a pre-processing step, where a data structure is computed that allows for fast querying of replacement paths, typically referred to as a Distance Sensitivity Oracle (DSO). While DSOs have been extensively studied in the theoretical computer science community, to the best of our knowledge this is the first work to construct DSOs using deep learning techniques. We show how to use deep learning to utilize a combinatorial structure of replacement paths. More specifically, we utilize the combinatorial structure of replacement paths as a concatenation of shortest paths and use deep learning to find the pivot nodes for stitching shortest paths into replacement paths.

Tobias Fiedler, Leon Hermann, Florian Müller et al.

The decoding of continuously spoken speech from neuronal activity has the potential to become an important clinical solution for paralyzed patients. Deep Learning Brain Computer Interfaces (BCIs) have recently successfully mapped neuronal activity to text contents in subjects who attempted to formulate speech. However, only small BCI datasets are available. In contrast, labeled data and pre-trained models for the closely related task of speech recognition from audio are widely available. One such model is Wav2Vec2 which has been trained in a self-supervised fashion to create meaningful representations of speech audio data. In this study, we show that patterns learned by Wav2Vec2 are transferable to brain data. Specifically, we replace its audio feature extractor with an untrained Brain Feature Extractor (BFE) model. We then execute full fine-tuning with pre-trained weights for Wav2Vec2, training ''from scratch'' without pre-trained weights as well as freezing a pre-trained Wav2Vec2 and training only the BFE each for 45 different BFE architectures. Across these experiments, the best run is from full fine-tuning with pre-trained weights, achieving a Character Error Rate (CER) of 18.54\%, outperforming the best training from scratch run by 20.46\% and that of frozen Wav2Vec2 training by 15.92\% percentage points. These results indicate that knowledge transfer from audio speech recognition to brain decoding is possible and significantly improves brain decoding performance for the same architectures. Related source code is available at https://github.com/tfiedlerdev/Wav2Vec2ForBrain.

Maximilian Böther, Otto Kißig, Martin Taraz et al.

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.

Michelle Döring, Jan Fehse, Tobias Friedrich et al.

The Vehicle Routing Problem (VRP) is a popular generalization of the Traveling Salesperson Problem. Instead of one salesperson traversing the entire weighted, undirected graph $G$, there are $k$ vehicles available to jointly cover the set of clients $C \subseteq V(G)$. Every vehicle must start at one of the depot vertices $D \subseteq V(G)$ and return to its start. Capacitated Vehicle Routing (CVRP) additionally restricts the route of each vehicle by limiting the number of clients it can cover, the distance it can travel, or both. In this work, we study the complexity of VRP and the three variants of CVRP for several parameterizations, in particular focusing on the treewidth of $G$. We present an FPT algorithm for VRP parameterized by treewidth. For CVRP, we prove paraNP- and $W[\cdot]$-hardness for various parameterizations, including treewidth, thereby rendering the existence of FPT algorithms unlikely. In turn, we provide an XP algorithm for CVRP when parameterized by both treewidth and the vehicle capacity.

Davide Bilò, Sarel Cohen, Tobias Friedrich et al.

Most networks are not static objects, but instead they change over time. This observation has sparked rigorous research on temporal graphs within the last years. In temporal graphs, we have a fixed set of nodes and the connections between them are only available at certain time steps. This gives rise to a plethora of algorithmic problems on such graphs, most prominently the problem of finding temporal spanners, i.e., the computation of subgraphs that guarantee all pairs reachability via temporal paths. To the best of our knowledge, only centralized approaches for the solution of this problem are known. However, many real-world networks are not shaped by a central designer but instead they emerge and evolve by the interaction of many strategic agents. This observation is the driving force of the recent intensive research on game-theoretic network formation models. In this work we bring together these two recent research directions: temporal graphs and game-theoretic network formation. As a first step into this new realm, we focus on a simplified setting where a complete temporal host graph is given and the agents, corresponding to its nodes, selfishly create incident edges to ensure that they can reach all other nodes via temporal paths in the created network. This yields temporal spanners as equilibria of our game. We prove results on the convergence to and the existence of equilibrium networks, on the complexity of finding best agent strategies, and on the quality of the equilibria. By taking these first important steps, we uncover challenging open problems that call for an in-depth exploration of the creation of temporal graphs by strategic agents.

Francesco Quinzan, Rajiv Khanna, Moshik Hershcovitch et al.

We study the fundamental problem of selecting optimal features for model construction. This problem is computationally challenging on large datasets, even with the use of greedy algorithm variants. To address this challenge, we extend the adaptive query model, recently proposed for the greedy forward selection for submodular functions, to the faster paradigm of Orthogonal Matching Pursuit for non-submodular functions. The proposed algorithm achieves exponentially fast parallel run time in the adaptive query model, scaling much better than prior work. Furthermore, our extension allows the use of downward-closed constraints, which can be used to encode certain fairness criteria into the feature selection process. We prove strong approximation guarantees for the algorithm based on standard assumptions. These guarantees are applicable to many parametric models, including Generalized Linear Models. Finally, we demonstrate empirically that the proposed algorithm competes favorably with state-of-the-art techniques for feature selection, on real-world and synthetic datasets.

Sebastian Angrick, Ben Bals, Niko Hastrich et al.

Human lives are increasingly influenced by algorithms, which therefore need to meet higher standards not only in accuracy but also with respect to explainability. This is especially true for high-stakes areas such as real estate valuation. Unfortunately, the methods applied there often exhibit a trade-off between accuracy and explainability. One explainable approach is case-based reasoning (CBR), where each decision is supported by specific previous cases. However, such methods can be wanting in accuracy. The unexplainable machine learning approaches are often observed to provide higher accuracy but are not scrutable in their decision-making. In this paper, we apply evolutionary algorithms (EAs) to CBR predictors in order to improve their performance. In particular, we deploy EAs to the similarity functions (used in CBR to find comparable cases), which are fitted to the data set at hand. As a consequence, we achieve higher accuracy than state-of-the-art deep neural networks (DNNs), while keeping interpretability and explainability. These results stem from our empirical evaluation on a large data set of real estate offers where we compare known similarity functions, their EA-improved counterparts, and DNNs. Surprisingly, DNNs are only on par with standard CBR techniques. However, using EA-learned similarity functions does yield an improved performance.

Francesco Quinzan, Vanja Doskoč, Andreas Göbel et al.

Several large-scale machine learning tasks, such as data summarization, can be approached by maximizing functions that satisfy submodularity. These optimization problems often involve complex side constraints, imposed by the underlying application. In this paper, we develop an algorithm with poly-logarithmic adaptivity for non-monotone submodular maximization under general side constraints. The adaptive complexity of a problem is the minimal number of sequential rounds required to achieve the objective. Our algorithm is suitable to maximize a non-monotone submodular function under a $p$-system side constraint, and it achieves a $(p + O(\sqrt{p}))$-approximation for this problem, after only poly-logarithmic adaptive rounds and polynomial queries to the valuation oracle function. Furthermore, our algorithm achieves a $(p + O(1))$-approximation when the given side constraint is a $p$-extendible system. This algorithm yields an exponential speed-up, with respect to the adaptivity, over any other known constant-factor approximation algorithm for this problem. It also competes with previous known results in terms of the query complexity. We perform various experiments on various real-world applications. We find that, in comparison with commonly used heuristics, our algorithm performs better on these instances.

Felix Mujkanovic, Vanja Doskoč, Martin Schirneck et al.

Modern time series classifiers display impressive predictive capabilities, yet their decision-making processes mostly remain black boxes to the user. At the same time, model-agnostic explainers, such as the recently proposed SHAP, promise to make the predictions of machine learning models interpretable, provided there are well-designed domain mappings. We bring both worlds together in our timeXplain framework, extending the reach of explainable artificial intelligence to time series classification and value prediction. We present novel domain mappings for the time domain, frequency domain, and time series statistics and analyze their explicative power as well as their limits. We employ a novel evaluation metric to experimentally compare timeXplain to several model-specific explanation approaches for state-of-the-art time series classifiers.

Vanja Doskoč, Tobias Friedrich, Andreas Göbel et al.

We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature. In contrast to other heuristics, this requires no problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size. In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings. Additionally, we study this problem in a dynamic setting, by which knapsacks change during the optimization process. We modify our greedy algorithm to avoid a complete restart at each constraint update. This modification retains the approximation guarantees of the static case. We evaluate our results experimentally on a video summarization and sensor placement task. We show that our proposed algorithm competes with the state-of-the-art in static settings. Furthermore, we show that in dynamic settings with tight computational time budget, our modified greedy yields significant improvements over starting the greedy from scratch, in terms of the solution quality achieved.

Álvaro Parra, Tat-Jun Chin, Frank Neumann et al.

A popular paradigm for 3D point cloud registration is by extracting 3D keypoint correspondences, then estimating the registration function from the correspondences using a robust algorithm. However, many existing 3D keypoint techniques tend to produce large proportions of erroneous correspondences or outliers, which significantly increases the cost of robust estimation. An alternative approach is to directly search for the subset of correspondences that are pairwise consistent, without optimising the registration function. This gives rise to the combinatorial problem of matching with pairwise constraints. In this paper, we propose a very efficient maximum clique algorithm to solve matching with pairwise constraints. Our technique combines tree searching with efficient bounding and pruning based on graph colouring. We demonstrate that, despite the theoretical intractability, many real problem instances can be solved exactly and quickly (seconds to minutes) with our algorithm, which makes our approach an excellent alternative to standard robust techniques for 3D registration.

Vahid Roostapour, Aneta Neumann, Frank Neumann et al.

We consider the subset selection problem for function $f$ with constraint bound $B$ that changes over time. Within the area of submodular optimization, various greedy approaches are commonly used. For dynamic environments we observe that the adaptive variants of these greedy approaches are not able to maintain their approximation quality. Investigating the recently introduced POMC Pareto optimization approach, we show that this algorithm efficiently computes a $φ= (α_f/2)(1-\frac{1}{e^{α_f}})$-approximation, where $α_f$ is the submodularity ratio of $f$, for each possible constraint bound $b \leq B$. Furthermore, we show that POMC is able to adapt its set of solutions quickly in the case that $B$ increases. Our experimental investigations for the influence maximization in social networks show the advantage of POMC over generalized greedy algorithms. We also consider EAMC, a new evolutionary algorithm with polynomial expected time guarantee to maintain $φ$ approximation ratio, and NSGA-II with two different population sizes as advanced multi-objective optimization algorithm, to demonstrate their challenges in optimizing the maximum coverage problem. Our empirical analysis shows that, within the same number of evaluations, POMC is able to perform as good as NSGA-II under linear constraint, while EAMC performs significantly worse than all considered algorithms in most cases.

Ankit Chauhan, Tobias Friedrich, Francesco Quinzan

It has been observed that many complex real-world networks have certain properties, such as a high clustering coefficient, a low diameter, and a power-law degree distribution. A network with a power-law degree distribution is known as scale-free network. In order to study these networks, various random graph models have been proposed, e.g. Preferential Attachment, Chung-Lu, or Hyperbolic. We look at the interplay between the power-law degree distribution and the run time of optimization techniques for well known combinatorial problems. We observe that on scale-free networks, simple evolutionary algorithms (EAs) quickly reach a constant-factor approximation ratio on common covering problems We prove that the single-objective (1+1)EA reaches a constant-factor approximation ratio on the Minimum Dominating Set problem, the Minimum Vertex Cover problem, the Minimum Connected Dominating Set problem, and the Maximum Independent Set problem in expected polynomial number of calls to the fitness function. Furthermore, we prove that the multi-objective GSEMO algorithm reaches a better approximation ratio than the (1+1)EA on those problems, within polynomial fitness evaluations.

Tobias Friedrich, Timo Kötzing, Markus Wagner

A common strategy for improving optimization algorithms is to restart the algorithm when it is believed to be trapped in an inferior part of the search space. However, while specific restart strategies have been developed for specific problems (and specific algorithms), restarts are typically not regarded as a general tool to speed up an optimization algorithm. In fact, many optimization algorithms do not employ restarts at all. Recently, "bet-and-run" was introduced in the context of mixed-integer programming, where first a number of short runs with randomized initial conditions is made, and then the most promising run of these is continued. In this article, we consider two classical NP-complete combinatorial optimization problems, traveling salesperson and minimum vertex cover, and study the effectiveness of different bet-and-run strategies. In particular, our restart strategies do not take any problem knowledge into account, nor are tailored to the optimization algorithm. Therefore, they can be used off-the-shelf. We observe that state-of-the-art solvers for these problems can benefit significantly from restarts on standard benchmark instances.

Duc-Cuong Dang, Tobias Friedrich, Timo Kötzing et al.

Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the ($μ$+1) GA and the $\text{Jump}_k$ test function. We show that the interplay of crossover and mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order $Ω(n/\log n)$ compared to mutation-only algorithms like (1+1) EA. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to speedups of order $Ω(n)$. We also compare seven commonly used diversity mechanisms and evaluate their impact on runtime bounds for the ($μ$+1) GA. All previous results in this context only hold for unrealistically low crossover probability $p_c=O(k/n)$, while we give analyses for the setting of constant $p_c < 1$ in all but one case. For the typical case of constant $k > 2$ and constant $p_c$, we can compare the resulting expected runtimes for different diversity mechanisms assuming an optimal choice of $μ$: $O(n^{k-1})$ for duplicate elimination/minim., $O(n^2\log n)$ for maximising the convex hull, $O(n\log n)$ for deterministic crowding (assuming $p_c = k/n$), $O(n\log n)$ for maximising Hamming distance, $O(n\log n)$ for fitness sharing, $O(n\log n)$ for single-receiver island model. This proves a sizeable advantage of all variants of the ($μ$+1) GA compared to (1+1) EA, which requires time $Θ(n^k)$. Experiments complement our theoretical findings and further highlight the benefits of crossover and diversity on $\text{Jump}_k$.

Tobias Friedrich, Timo Kötzing, Martin Krejca et al.

The benefit of sexual recombination is one of the most fundamental questions both in population genetics and evolutionary computation. It is widely believed that recombination helps solving difficult optimization problems. We present the first result, which rigorously proves that it is beneficial to use sexual recombination in an uncertain environment with a noisy fitness function. For this, we model sexual recombination with a simple estimation of distribution algorithm called the Compact Genetic Algorithm (cGA), which we compare with the classical $μ+1$ EA. For a simple noisy fitness function with additive Gaussian posterior noise $\mathcal{N}(0,σ^2)$, we prove that the mutation-only $μ+1$ EA typically cannot handle noise in polynomial time for $σ^2$ large enough while the cGA runs in polynomial time as long as the population size is not too small. This shows that in this uncertain environment sexual recombination is provably beneficial. We observe the same behavior in a small empirical study.

Tobias Friedrich, Markus Wagner

Most experimental studies initialize the population of evolutionary algorithms with random genotypes. In practice, however, optimizers are typically seeded with good candidate solutions either previously known or created according to some problem-specific method. This "seeding" has been studied extensively for single-objective problems. For multi-objective problems, however, very little literature is available on the approaches to seeding and their individual benefits and disadvantages. In this article, we are trying to narrow this gap via a comprehensive computational study on common real-valued test functions. We investigate the effect of two seeding techniques for five algorithms on 48 optimization problems with 2, 3, 4, 6, and 8 objectives. We observe that some functions (e.g., DTLZ4 and the LZ family) benefit significantly from seeding, while others (e.g., WFG) profit less. The advantage of seeding also depends on the examined algorithm.

Tobias Friedrich, Frank Neumann, Christian Thyssen

Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing μ points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations.