arXiv subjects

Tobias Friedrich

Publications and source records attributed to Tobias Friedrich.

At least 19 recordsLinked to original sources

arXiv2026

Fault-Tolerant ST-Diameter Oracles

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.

cs.DS↗

arXiv2025

Polymer Dynamics via Cliques: New Conditions for Approximations

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition -- the clique dynamics condition -- , which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least $2^{1/α}$ for the hard-core model on bipartite $α$-expanders.

math.PR↗

arXiv2025

Optimal Padded Decomposition For Bounded Treewidth Graphs

A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Δ$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,γΔ)$ is entirely contained in the cluster containing $v$ is at least $e^{-βγ}$ for every $γ\in [0,δ]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $β$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $β= Θ(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $β= O(\log r)$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.

cs.DS↗

arXiv2025

Parameterized Complexity of Vehicle Routing

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.

cs.CC↗

arXiv2025

Binary $k$-Center with Missing Entries: Structure Leads to Tractability

$\kC$ clustering is a fundamental classification problem, where the task is to categorize the given collection of entities into $k$ clusters and come up with a representative for each cluster, so that the maximum distance between an entity and its representative is minimized. In this work, we focus on the setting where the entities are represented by binary vectors with missing entries, which model incomplete categorical data. This version of the problem has wide applications, from predictive analytics to bioinformatics. Our main finding is that the problem, which is notoriously hard from the classical complexity viewpoint, becomes tractable as soon as the known entries are sparse and exhibit a certain structure. Formally, we show fixed-parameter tractable algorithms for the parameters vertex cover, fracture number, and treewidth of the row-column graph, which encodes the positions of the known entries of the matrix. Additionally, we tie the complexity of the 1-cluster variant of the problem, which is famous under the name Closest String, to the complexity of solving integer linear programs with few constraints. This implies, in particular, that improving upon the running times of our algorithms would lead to more efficient algorithms for integer linear programming in general.

cs.DS↗

arXiv2025

Strategic Network Creation for Enabling Greedy Routing

Today we rely on networks that are created and maintained by smart devices. For such networks, there is no governing central authority but instead the network structure is shaped by the decisions of selfish intelligent agents. A key property of such communication networks is that they should be easy to navigate for routing data. For this, a common approach is greedy routing, where every device simply routes data to a neighbor that is closer to the respective destination. Networks of intelligent agents can be analyzed via a game-theoretic approach and in the last decades many variants of network creation games have been proposed and analyzed. In this paper we present the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in our model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled. Besides this, the agents optimize their connection quality within the created network by aiming for greedy routing paths with low stretch. For our model, we analyze the existence of (approximate)-equilibria and the computational hardness in different underlying metric spaces. E.g., we characterize the set of equilibria in 1-2-metrics and tree metrics and show that Nash equilibria always exist. For Euclidean space, the setting which is most relevant in practice, we prove that equilibria are not guaranteed to exist but that the well-known $Θ$-graph construction yields networks having a low stretch that are game-theoretically almost stable. For general metric spaces, we show that approximate equilibria exist where the approximation factor depends on the cost of maintaining any link.

cs.GT↗

arXiv2025

Teaching Wav2Vec2 the Language of the Brain

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.

cs.LG↗

arXiv2024

Efficient Fault-Tolerant Search by Fast Indexing of Subnetworks

We design sensitivity oracles for error-prone networks. For a network problem $Π$, the data structure preprocesses a network $G=(V,E)$ and sensitivity parameter $f$ such that, for any set $F\subseteq V\cup E$ of up to $f$ link or node failures, it can report a solution for $Π$ in $G{-}F$. We study three network problems $Π$. $L$-Hop Shortest Path: Given $s,t \in V$, is there a shortest $s$-$t$-path in $G-F$ with at most $L$ links? $k$-Path: Does $G-F$ contain a simple path with $k$ links? $k$-Clique: Does $G-F$ contain a clique of $k$ nodes? Our main technical contribution is a new construction of $(L,f)$-replacement path coverings ($(L,f)$-RPC) in the parameter realm where $f = o(\log L)$. An $(L,f)$-RPC is a family $\mathcal{G}$ of subnetworks of $G$ which, for every $F \subseteq E$ with $|F| \le f$, contain a subfamily $\mathcal{G}_F \subseteq \mathcal{G}$ such that (i) no subnetwork in $\mathcal{G}_F$ contains a link of $F$ and (ii) for each $s,t \in V$, if $G-F$ contains a shortest $s$-$t$-path with at most $L$ links, then some subnetwork in $\mathcal{G}_F$ retains at least one such path. Our $(L, f)$-RPC has almost the same size as the one by Weimann and Yuster [ACM TALG 2013] but it improves the time to query $\mathcal{G}_F$ from $\widetilde{O}(f^2L^f)$ to $\widetilde{O}(f^{\frac{5}{2}} L^{o(1)})$. It also improves over the size and query time of the $(L,f)$-RPC by Karthik and Parter [SODA 2021] by nearly a factor of $L$. We then derive oracles for $L$-Hop Shortest Path, $k$-Path, and $k$-Clique from this. Notably, our solution for $k$-Path improves the query time of the one by Bilò, et al. [ITCS 2022] for $f=o(\log k)$.

cs.DS↗

arXiv2024

From Market Saturation to Social Reinforcement: Understanding the Impact of Non-Linearity in Information Diffusion Models

Diffusion of information in networks is at the core of many problems in AI. Common examples include the spread of ideas and rumors as well as marketing campaigns. Typically, information diffuses at a non-linear rate, for example, if markets become saturated or if users of social networks reinforce each other's opinions. Despite these characteristics, this area has seen little research, compared to the vast amount of results for linear models, which exhibit less complex dynamics. Especially, when considering the possibility of re-infection, no fully rigorous guarantees exist so far. We address this shortcoming by studying a very general non-linear diffusion model that captures saturation as well as reinforcement. More precisely, we consider a variant of the SIS model in which vertices get infected at a rate that scales polynomially in the number of their infected neighbors, weighted by an infection coefficient $λ$. We give the first fully rigorous results for thresholds of $λ$ at which the expected survival time becomes super-polynomial. For cliques we show that when the infection rate scales sub-linearly, the threshold only shifts by a poly-logarithmic factor, compared to the standard SIS model. In contrast, super-linear scaling changes the process considerably and shifts the threshold by a polynomial term. For stars, sub-linear and super-linear scaling behave similar and both shift the threshold by a polynomial factor. Our bounds are almost tight, as they are only apart by at most a poly-logarithmic factor from the lower thresholds, at which the expected survival time is logarithmic.

math.PR↗

arXiv2024

Combining Crown Structures for Vulnerability Measures

Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices required to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al.~[7] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from $p^3$ to $p^2$, and of wVI from $p^3$ to $3(p^2 + p^{1.5} p_{\ell})$, where $p_{\ell} < p$ represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from $\mathcal{O}(k^2W + kW^2)$ to $3μ(k + \sqrtμW)$, where $μ= \max(k,W)$. We also give a combinatorial algorithm that provides a $2kW$ vertex kernel in FPT-runtime when parameterized by $r$, where $r \leq k$ is the size of a maximum $(W+1)$-packing. We further show that the algorithm computing the $2kW$ vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when $W=1$, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a $2k$ vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.

cs.DS↗

arXiv2024

Temporal Network Creation Games: The Impact of Non-Locality and Terminals

We live in a world full of networks where our economy, our communication, and even our social life crucially depends on them. These networks typically emerge from the interaction of many entities, which is why researchers study agent-based models of network formation. While traditionally static networks with a fixed set of links were considered, a recent stream of works focuses on networks whose behavior may change over time. In particular, Bilò et al. (IJCAI 2023) recently introduced a game-theoretic network formation model that embeds temporal aspects in networks. More precisely, a network is formed by selfish agents corresponding to nodes in a given host network with edges having labels denoting their availability over time. Each agent strategically selects local, i.e., incident, edges to ensure temporal reachability towards everyone at low cost. In this work we set out to explore the impact of two novel conceptual features: agents are no longer restricted to creating incident edges, called the global setting, and agents might only want to ensure that they can reach a subset of the other nodes, called the terminal model. For both, we study the existence, structure, and quality of equilibrium networks. For the terminal model, we prove that many core properties crucially depend on the number of terminals. We also develop a novel tool that allows translating equilibrium constructions from the non-terminal model to the terminal model. For the global setting, we show the surprising result that equilibria in the global and the local model are incomparable and we establish a high lower bound on the Price of Anarchy of the global setting that matches the upper bound of the local model. This shows the counter-intuitive fact that allowing agents more flexibility in edge creation does not improve the quality of equilibrium networks.

cs.GT↗

arXiv2024

Improved Distance (Sensitivity) Oracles with Subquadratic Space

A distance oracle (DO) with stretch $(α, β)$ for a graph $G$ is a data structure that, when queried with vertices $s$ and $t$, returns a value $\widehat{d}(s,t)$ such that $d(s,t) \le \widehat{d}(s,t) \le α\cdot d(s,t) + β$. An $f$-edge fault-tolerant distance sensitivity oracle ($f$-DSO) additionally receives a set $F$ of up to $f$ edges and estimates the $s$-$t$-distance in $G{-}F$. Our first contribution is a new distance oracle with subquadratic space for undirected graphs. Introducing a small additive stretch $β> 0$ allows us to make the multiplicative stretch $α$ arbitrarily small. This sidesteps a known lower bound of $α\ge 3$ (for $β= 0$ and subquadratic space) [Thorup & Zwick, JACM 2005]. We present a DO for graphs with edge weights in $[0,W]$ that, for any positive integer $t$ and any $c \in (0, \ell/2]$, has stretch $(1{+}\frac{1}{\ell}, 2W)$, space $\widetilde{O}(n^{2-\frac{c}{t}})$, and query time $O(n^c)$. These are the first subquadratic-space DOs with $(1+ε, O(1))$-stretch generalizing Agarwal and Godfrey's results for sparse graphs [SODA 2013] to general undirected graphs. Our second contribution is a framework that turns a $(α,β)$-stretch DO for unweighted graphs into an $(α(1{+}\varepsilon),β)$-stretch $f$-DSO with sensitivity $f = o(\log(n)/\log\log n)$ and retains subquadratic space. This generalizes a result by Bilò, Chechik, Choudhary, Cohen, Friedrich, Krogmann, and Schirneck [STOC 2023, TheoretiCS 2024] for the special case of stretch $(3,0)$ and $f = O(1)$. By combining the framework with our new distance oracle, we obtain an $f$-DSO that, for any $γ\in (0, (\ell{+}1)/2]$, has stretch $((1{+}\frac{1}{\ell}) (1{+}\varepsilon), 2)$, space $n^{ 2- \fracγ{(\ell+1)(f+1)} + o(1)}/\varepsilon^{f+2}$, and query time $\widetilde{O}(n^γ /{\varepsilon}^2)$.

cs.DS↗

arXiv2024

Real-World Networks are Low-Dimensional: Theoretical and Practical Assessment

Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are computationally heavy and lack theoretical foundation. On the other hand, theoretical approaches do not apply to graphs with heterogeneous degree distributions, which is often the case for complex real-world networks. To address these drawbacks, we consider geometric inhomogeneous random graphs (GIRGs) as a random graph model, which captures a variety of properties observed in practice. Our first result shows that the clustering coefficient of GIRGs scales inverse exponentially with respect to the number of dimensions, when the latter is at most logarithmic in $n$. This gives a first theoretical explanation for the low dimensionality of real-world networks as observed by Almagro et al. in 2022. We further use these insights to derive a linear-time algorithm for determining the dimensionality of a given GIRG and prove that our algorithm returns the correct number of dimensions with high probability GIRG. Our algorithm bridges the gap between theory and practice, as it not only comes with a rigorous proof of correctness but also yields results comparable to that of prior empirical approaches, as indicated by our experiments on real-world instances.

cs.SI↗

arXiv2024

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

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.

cs.DM↗

arXiv2024

How to Reduce Temporal Cliques to Find Sparse Spanners

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

cs.DM↗

arXiv2024

Winning Through Simplicity: Autonomous Car Design for Formula Student

This paper presents the design of an autonomous race car that is self-designed, self-developed, and self-built by the Elefant Racing team at the University of Bayreuth. The system is created to compete in the Formula Student Driverless competition. Its primary focus is on the Acceleration track, a straight 75-meter-long course, and the Skidpad track, which comprises two circles forming an eight. Additionally, it is experimentally capable of competing in the Autocross and Trackdrive events, which feature tracks with previously unknown straights and curves. The paper details the hardware, software and sensor setup employed during the 2020/2021 season. Despite being developed by a small team with limited computer science expertise, the design won the Formula Student East Engineering Design award. Emphasizing simplicity and efficiency, the team employed streamlined techniques to achieve their success.

cs.RO↗

arXiv2024

Approximate Distance Sensitivity Oracles in Subquadratic Space

An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $σ\ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $σ$-approximation of the $s$-$t$-distance in $G-F$. We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $σ\ge 3$. We present, for any constant $f \ge 1$ and $α\in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^α/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$. Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$.

cs.DS↗

arXiv2024

Robust Parameter Fitting to Realistic Network Models via Iterative Stochastic Approximation

Random graph models are widely used to understand network properties and graph algorithms. Key to such analyses are the different parameters of each model, which affect various network features, such as its size, clustering, or degree distribution. The exact effect of the parameters on these features is not well understood, mainly because we lack tools to thoroughly investigate this relation. Moreover, the parameters cannot be considered in isolation, as changing one affects multiple features. Existing approaches for finding the best model parameters of desired features, such as a grid search or estimating the parameter-feature relations, are not well suited, as they are inaccurate or computationally expensive. We introduce an efficient iterative fitting method, named ParFit, that finds parameters using only a few network samples, based on the Robbins-Monro algorithm. We test ParFit on three well-known graph models, namely Erdős-Rényi, Chung-Lu, and geometric inhomogeneous random graphs, as well as on real-world networks, including web networks. We find that ParFit performs well in terms of quality and running time across most parameter configurations.

cs.SI↗