arXiv subjects

Tobias Friedrich

Publications and source records attributed to Tobias Friedrich.

At least 55 records · Page 3Linked to original sources

arXiv2022

Strongly Hyperbolic Unit Disk Graphs

The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.

cs.DS↗

arXiv2022

Theoretical Study of Optimizing Rugged Landscapes with the cGA

Estimation of distribution algorithms (EDAs) provide a distribution - based approach for optimization which adapts its probability distribution during the run of the algorithm. We contribute to the theoretical understanding of EDAs and point out that their distribution approach makes them more suitable to deal with rugged fitness landscapes than classical local search algorithms. Concretely, we make the OneMax function rugged by adding noise to each fitness value. The cGA can nevertheless find solutions with n(1 - ε) many 1s, even for high variance of noise. In contrast to this, RLS and the (1+1) EA, with high probability, only find solutions with n(1/2+o(1)) many 1s, even for noise with small variance.

cs.NE↗

arXiv2022

Approximate Max-Flow Min-Multicut Theorem for Graphs of Bounded Treewidth

We prove an approximate max-multiflow min-multicut theorem for bounded treewidth graphs. In particular, we show the following: Given a treewidth-$r$ graph, there exists a (fractional) multicommodity flow of value $f$, and a multicut of capacity $c$ such that $ f \leq c \leq \mathcal{O}(\ln (r+1)) \cdot f$. It is well known that the multiflow-multicut gap on an $r$-vertex (constant degree) expander graph can be $Ω(\ln r)$, and hence our result is tight up to constant factors. Our proof is constructive, and we also obtain a polynomial time $\mathcal{O}(\ln (r+1))$-approximation algorithm for the minimum multicut problem on treewidth-$r$ graphs. Our algorithm proceeds by rounding the optimal fractional solution to the natural linear programming relaxation of the multicut problem. We introduce novel modifications to the well-known region growing algorithm to facilitate the rounding while guaranteeing at most a logarithmic factor loss in the treewidth.

cs.DS↗

arXiv2022

Analysis of the survival time of the SIRS process via expansion

We study the SIRS process, a continuous-time Markov chain modeling the spread of infections on graphs. In this model, vertices are either susceptible, infected, or recovered. Each infected vertex becomes recovered at rate 1 and infects each of its susceptible neighbors independently at rate $λ$, and each recovered vertex becomes susceptible at a rate $\varrho$, which we assume to be independent of the graph size. A central quantity of the SIRS process is the time until no vertex is infected, known as the survival time. Surprisingly though, rigorous theoretical results exist only for the related SIS model so far. We address this imbalance by conducting theoretical analyses of the SIRS process via their expansion properties. We prove that the expected survival time of the SIRS process on stars is at most polynomial in the graph size for any value of $λ$. This behavior is fundamentally different from the SIS process, where the expected survival time is exponential already for small infection rates. Our main result is an exponential lower bound of the expected survival time of the SIRS process on expander graphs. Specifically, we show that on expander graphs $G$ with $n$ vertices, degree close to $d$, and sufficiently small spectral expansion, the SIRS process has expected survival time at least exponential in $n$ when $λ\geq c/d$ for a constant $c > 1$. Previous results on the SIS process show that this bound is almost tight. Additionally, our result holds even if $G$ is a subgraph. Notably, our result implies an almost-tight threshold for Erdos-Rényi graphs and a regime of exponential survival time for hyperbolic random graphs. The proof of our main result draws inspiration from Lyapunov functions used in mean-field theory to devise a two-dimensional potential function and applying a negative-drift theorem to show that the expected survival time is exponential.

math.PR↗

arXiv2022

The Satisfiability Threshold for Non-Uniform Random 2-SAT

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of non-uniform random $2$-SAT with $n$ variables and $m$ clauses and with an arbitrary probability distribution $(p_i)_{i\in[n]}$ with $p_1 \ge p_2 \ge \ldots \ge p_n > 0$ over the n variables. We show for $p_1^2=Θ(\sum_{i=1}^n p_i^2)$ that the asymptotic satisfiability threshold is at $m=Θ( (1-\sum_{i=1}^n p_i^2)/(p_1\cdot(\sum_{i=2}^n p_i^2)^{1/2}) )$ and that it is coarse. For $p_1^2=o(\sum_{i=1}^n p_i^2)$ we show that there is a sharp satisfiability threshold at $m=(\sum_{i=1}^n p_i^2)^{-1}$. This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].

cs.DM↗

arXiv2022

On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

We study the problem of counting the number of homomorphisms from an input graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and Jerrum [ToC'15] and its complexity is subject to a growing series of research articles, e.g. the work of Focke, Goldberg, Roth, and Zivný [SIDMA'21] and the work of Bulatov and Kazeminia [STOC'22], subsequent to this article's conference version. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph $\bar{H}$ collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite ($K_{3,3}\backslash\{e\}$, ${domino}$)-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with $p$ equal to $2$, this establishes new results.

cs.CC↗

arXiv2022

Social Distancing Network Creation

During a pandemic people have to find a trade-off between meeting others and staying safely at home. While meeting others is pleasant, it also increases the risk of infection. We consider this dilemma by introducing a game-theoretic network creation model in which selfish agents can form bilateral connections. They benefit from network neighbors, but at the same time, they want to maximize their distance to all other agents. This models the inherent conflict that social distancing rules impose on the behavior of selfish agents in a social network. Besides addressing this familiar issue, our model can be seen as the inverse to the well-studied Network Creation Game by Fabrikant et al. [PODC 2003] where agents aim at being as central as possible in the created network. Thus, our work is in-line with studies that compare minimization problems with their maximization versions. We look at two variants of network creation governed by social distancing. In the first variant, there are no restrictions on the connections being formed. We characterize optimal and equilibrium networks, and we derive asymptotically tight bounds on the Price of Anarchy and Price of Stability. The second variant is the model's generalization that allows restrictions on the connections that can be formed. As our main result, we prove that Swap-Maximal Routing-Cost Spanning Trees, an efficiently computable weaker variant of Maximum Routing-Cost Spanning Trees, actually resemble equilibria for a significant range of the parameter space. Moreover, we give almost tight bounds on the Price of Anarchy and Price of Stability. These results imply that, compared the well-studied inverse models, under social distancing the agents' selfish behavior has a significantly stronger impact on the quality of the equilibria, i.e., allowing socially much worse stable states.

cs.GT↗

arXiv2022

Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances

We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs. Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen [ICALP 2019] as well as Karthik and Parter [SODA 2021], we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter and eccentricity oracles and then show its versatility by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren [JCSS 2022] and the Single-Source Replacement Path algorithm of Chechik and Magen [ICALP 2020]. This way, we obtain the first deterministic distance sensitivity oracle with subcubic preprocessing time.

cs.DS↗

arXiv2022

Towards Explainable Real Estate Valuation via Evolutionary Algorithms

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.

cs.NE↗

arXiv2022

Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry

A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is $\mathcal {\tilde O}(n^{2 - 1/α} + n^{1/(2α)} + δ_{\max})$ with high probability, where $α\in (0.5, 1)$ controls the power-law exponent of the degree distribution, and $δ_{\max}$ is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.

cs.DS↗

arXiv2022

Analysis of a Gray-Box Operator for Vertex Cover

Combinatorial optimization problems are a prominent application area of evolutionary algorithms, where the (1+1) EA is one of the most investigated. We extend this algorithm by introducing some problem knowledge with a specialized mutation operator which works under the assumption that the number of 1s of a solution is critical, as frequently happens in combinatorial optimization. This slight modification increases the chance to correct wrongly placed bits while preserving the simplicity and problem independence of the (1+1) EA. As an application of our algorithm we examine the vertex cover problem on certain instances, where we show that it leads to asymptotically better runtimes and even finds with higher probability optimal solutions in comparison with the usual (1+1) EA. Precisely, we compare the performance of both algorithms on paths and on complete bipartite graphs of size $n$. Regarding the path we prove that, for a particular initial configuration, the \alg1+1 takes in expectation $Θ(n^4)$ iterations while the modification reduces this to $Θ(n^3)$, and present experimental evidence that such a configuration is reached. Concerning the complete bipartite graph our modification finds the optimum in polynomial time with probability $1-1/2^{Ω(n^ξ)}$ for every positive constant $ξ< 1$, which improves the known probability of $1-1/\text{poly}(n)$ for the (1+1) EA..

math.CO↗

arXiv2022

Algorithms for hard-constraint point processes via discretization

We study algorithmic applications of a natural discretization for the hard-sphere model and the Widom-Rowlinson model in a region $\mathbb{V}\subset\mathbb{R}^d$. These models are used in statistical physics to describe mixtures of one or multiple particle types subjected to hard-core interactions. For each type, particles follow a Poisson point process with a type specific activity parameter (fugacity). The Gibbs distribution is characterized by the mixture of these point processes conditioned that no two particles are closer than a type-dependent distance threshold. A key part in better understanding the Gibbs distribution is its normalizing constant, called partition function. We give sufficient conditions that the partition function of a discrete hard-core model on a geometric graph based on a point set $X \subset \mathbb{V}$ closely approximates those of such continuous models. Previously, this was only shown for the hard-sphere model on cubic regions $\mathbb{V}=[0, \ell)^d$ when $X$ is exponential in the volume of the region $ν(\mathbb{V})$, limiting algorithmic applications. In the same setting, our refined analysis only requires a quadratic number of points, which we argue to be tight. We use our improved discretization results to approximate the partition functions of the hard-sphere model and the Widom-Rowlinson efficiently in $ν(\mathbb{V})$. For the hard-sphere model, we obtain the first quasi-polynomial deterministic approximation algorithm for the entire fugacity regime for which, so far, only randomized approximations are known. Furthermore, we simplify a recently introduced fully polynomial randomized approximation algorithm. Similarly, we obtain the best known deterministic and randomized approximation bounds for the Widom-Rowlinson model. Moreover, we obtain approximate sampling algorithms for the respective spin systems within the same fugacity regimes.

cs.DS↗

arXiv2022

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

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.

cs.LG↗

arXiv2021

Fixed-Parameter Sensitivity Oracles

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity $f$ for an FPT problem $Π$ on a graph $G$ with parameter $k$ preprocesses $G$ in time $O(g(f,k) \cdot \textsf{poly}(n))$. When queried with a set $F$ of at most $f$ edges of $G$, the oracle reports the answer to the $Π$-with the same parameter $k$-on the graph $G-F$, i.e., $G$ deprived of $F$. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on $G-F$ from scratch. We mainly design sensitivity oracles for the $k$-Path and the $k$-Vertex Cover problem. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we also introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source $s$ and parameters $f$ and $k$, to construct a polynomial-sized oracle that efficiently reports, for any target vertex $v$ and set $F$ of at most $f$ edges, whether the distance from $s$ to $v$ increases at most by an additive term of $k$ in $G-F$.

cs.DS↗

arXiv2021

The Impact of Heterogeneity and Geometry on the Proof Complexity of Random Satisfiability

Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random k-SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. A key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. As an additional technical contribution, we show a linear upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. This is in stark contrast to quadratic lower bounds for the worst case.

cs.CC↗

arXiv2021

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

The transversal hypergraph problem is the task of enumerating the minimal hitting sets of a hypergraph. It is a long-standing open question whether this can be done in output-polynomial time. For hypergraphs whose solutions have bounded size, Eiter and Gottlob [SICOMP 1995] gave an algorithm that runs in output-polynomial time, but whose space requirement also scales with the output size. We improve this to polynomial delay and polynomial space. More generally, we present an algorithm that on $n$-vertex, $m$-edge hypergraphs has delay $O(m^{k^*+1} n^2)$ and uses $O(mn)$ space, where $k^*$ is the maximum size of any minimal hitting set. Our algorithm is oblivious to $k^*$, a quantity that is hard to compute or even approximate. Central to our approach is the extension problem for minimal hitting sets, deciding for a set $X$ of vertices whether it is contained in any solution. With $|X|$ as parameter, we show that this is one of the first natural problems to be complete for the complexity class $W[3]$. We give an algorithm for the extension problem running in time $O(m^{|X|+1} n)$. We also prove a conditional lower bound under the Strong Exponential Time Hypothesis, showing that this is close to optimal. We apply our enumeration method to the discovery problem of minimal unique column combinations from data profiling. Our empirical evaluation suggests that the algorithm outperforms its worst-case guarantees on hypergraphs stemming from real-world databases.

cs.DS↗

arXiv2021

Pareto Optimization for Subset Selection with Dynamic Cost Constraints

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.

cs.DS↗

arXiv2021

Pareto optimal and popular house allocation with lower and upper quotas

In the house allocation problem with lower and upper quotas, we are given a set of applicants and a set of projects. Each applicant has a strictly ordered preference list over the projects, while the projects are equipped with a lower and an upper quota. A feasible matching assigns the applicants to the projects in such a way that a project is either matched to no applicant or to a number of applicants between its lower and upper quota. In this model we study two classic optimality concepts: Pareto optimality and popularity. We show that finding a popular matching is hard even if the maximum lower quota is 2 and that finding a perfect Pareto optimal matching, verifying Pareto optimality, and verifying popularity are all NP-complete even if the maximum lower quota is 3. We complement the last three negative results by showing that the problems become polynomial-time solvable when the maximum lower quota is 2, thereby answering two open questions of Cechlárová and Fleiner. Finally, we also study the parameterized complexity of all four mentioned problems.

cs.GT↗