507 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On stability of metric spaces and Kalton's property QQ

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    The first named author introduced the notion of upper stability for metric spaces as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property QQ does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces.Comment: 14 page

    Excluding Surfaces as Minors in Graphs

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    We introduce an annotated extension of treewidth that measures the contribution of a vertex set XX to the treewidth of a graph G.G. This notion provides a graph distance measure to some graph property P\mathcal{P}: A vertex set XX is a kk-treewidth modulator of GG to P\mathcal{P} if the treewidth of XX in GG is at most kk and its removal gives a graph in P.\mathcal{P}.This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: KkK_k-minor free graphs are those that admit tree-decompositions whose torsos have ckc_{k}-treewidth modulators to some surface of Euler-genus ck.c_{k}. This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that Σ\Sigma is some particular surface and define a ``surface extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum kk for which GG admits a tree-decomposition whose torsos have a kk-treewidth modulator to being embeddable in Σ.\Sigma.We identify a finite collection DΣ\mathfrak{D}_{\Sigma} of parametric graphs and prove that the minor-exclusion of the graphs in DΣ\mathfrak{D}_{\Sigma} precisely determines the asymptotic behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface Σ.\Sigma. It follows that the collection DΣ\mathfrak{D}_{\Sigma} bijectively corresponds to the ``surface obstructions'' for Σ,\Sigma, i.e., surfaces that are minimally non-contained in $\Sigma.

    Planar Disjoint Paths, Treewidth, and Kernels

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    In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of kk vertex pairs (si,ti)(s_i,t_i) and the task is to find kk pairwise vertex-disjoint paths such that the ii-th path connects sis_i to tit_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by kk unless coNP \subseteq NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by k+twk + tw, where twtw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to tw=kO(1)tw= k^{O(1)} under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time 2O(k2)nO(1)2^{O(k^2)}\cdot n^{O(1)}, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

    Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

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    Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in its (open) neighborhood. Typical examples of graphs with large proper conflict-free chromatic number include graphs with large chromatic number and bipartite graphs isomorphic to the 11-subdivision of graphs with large chromatic number. In this paper, we prove two rough converse statements that hold even in the list-coloring setting. The first is for sparse graphs: for every graph HH, there exists an integer cHc_H such that every graph with no subdivision of HH is (properly) conflict-free cHc_H-choosable. The second applies to dense graphs: every graph with large conflict-free choice number either contains a large complete graph as an odd minor or contains a bipartite induced subgraph that has large conflict-free choice number. These give two incomparable (partial) answers of a question of Caro, Petru\v{s}evski and \v{S}krekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about proper conflict-free coloring and odd coloring in the literature. Moreover, we prove that every graph with layered treewidth at most ww is (properly) conflict-free (8w1)(8w-1)-choosable. This result applies to (g,k)(g,k)-planar graphs, which are graphs whose coloring problems have attracted attention recently.Comment: Hickingbotham recently independently announced a paper (arXiv:2203.10402) proving a result similar to the ones in this paper. Please see the notes at the end of this paper for details. v2: add results for odd minors, which applies to graphs with unbounded degeneracy, and change the title of the pape

    Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

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    In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most O(1)O(1). Next, we provide an efficient O(logn)O(\log n)-approximation algorithm for finding the minimum ratio-cut of GG. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time O(logn)O(\log n)-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time O(logn)O\big(\sqrt{\log n}\big)-approximation algorithm for the minimum ratio-cut problem based on rounding 22\ell_2^2-metric embeddings.Comment: Comments and feedback welcom

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Modern data analytics in the cloud era

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    Cloud Computing ist die dominante Technologie des letzten Jahrzehnts. Die Benutzerfreundlichkeit der verwalteten Umgebung in Kombination mit einer nahezu unbegrenzten Menge an Ressourcen und einem nutzungsabhängigen Preismodell ermöglicht eine schnelle und kosteneffiziente Projektrealisierung für ein breites Nutzerspektrum. Cloud Computing verändert auch die Art und Weise wie Software entwickelt, bereitgestellt und genutzt wird. Diese Arbeit konzentriert sich auf Datenbanksysteme, die in der Cloud-Umgebung eingesetzt werden. Wir identifizieren drei Hauptinteraktionspunkte der Datenbank-Engine mit der Umgebung, die veränderte Anforderungen im Vergleich zu traditionellen On-Premise-Data-Warehouse-Lösungen aufweisen. Der erste Interaktionspunkt ist die Interaktion mit elastischen Ressourcen. Systeme in der Cloud sollten Elastizität unterstützen, um den Lastanforderungen zu entsprechen und dabei kosteneffizient zu sein. Wir stellen einen elastischen Skalierungsmechanismus für verteilte Datenbank-Engines vor, kombiniert mit einem Partitionsmanager, der einen Lastausgleich bietet und gleichzeitig die Neuzuweisung von Partitionen im Falle einer elastischen Skalierung minimiert. Darüber hinaus führen wir eine Strategie zum initialen Befüllen von Puffern ein, die es ermöglicht, skalierte Ressourcen unmittelbar nach der Skalierung auszunutzen. Cloudbasierte Systeme sind von fast überall aus zugänglich und verfügbar. Daten werden häufig von zahlreichen Endpunkten aus eingespeist, was sich von ETL-Pipelines in einer herkömmlichen Data-Warehouse-Lösung unterscheidet. Viele Benutzer verzichten auf die Definition von strikten Schemaanforderungen, um Transaktionsabbrüche aufgrund von Konflikten zu vermeiden oder um den Ladeprozess von Daten zu beschleunigen. Wir führen das Konzept der PatchIndexe ein, die die Definition von unscharfen Constraints ermöglichen. PatchIndexe verwalten Ausnahmen zu diesen Constraints, machen sie für die Optimierung und Ausführung von Anfragen nutzbar und bieten effiziente Unterstützung bei Datenaktualisierungen. Das Konzept kann auf beliebige Constraints angewendet werden und wir geben Beispiele für unscharfe Eindeutigkeits- und Sortierconstraints. Darüber hinaus zeigen wir, wie PatchIndexe genutzt werden können, um fortgeschrittene Constraints wie eine unscharfe Multi-Key-Partitionierung zu definieren, die eine robuste Anfrageperformance bei Workloads mit unterschiedlichen Partitionsanforderungen bietet. Der dritte Interaktionspunkt ist die Nutzerinteraktion. Datengetriebene Anwendungen haben sich in den letzten Jahren verändert. Neben den traditionellen SQL-Anfragen für Business Intelligence sind heute auch datenwissenschaftliche Anwendungen von großer Bedeutung. In diesen Fällen fungiert das Datenbanksystem oft nur als Datenlieferant, während der Rechenaufwand in dedizierten Data-Science- oder Machine-Learning-Umgebungen stattfindet. Wir verfolgen das Ziel, fortgeschrittene Analysen in Richtung der Datenbank-Engine zu verlagern und stellen das Grizzly-Framework als DataFrame-zu-SQL-Transpiler vor. Auf dieser Grundlage identifizieren wir benutzerdefinierte Funktionen (UDFs) und maschinelles Lernen (ML) als wichtige Aufgaben, die von einer tieferen Integration in die Datenbank-Engine profitieren würden. Daher untersuchen und bewerten wir Ansätze für die datenbankinterne Ausführung von Python-UDFs und datenbankinterne ML-Inferenz.Cloud computing has been the groundbreaking technology of the last decade. The ease-of-use of the managed environment in combination with nearly infinite amount of resources and a pay-per-use price model enables fast and cost-efficient project realization for a broad range of users. Cloud computing also changes the way software is designed, deployed and used. This thesis focuses on database systems deployed in the cloud environment. We identify three major interaction points of the database engine with the environment that show changed requirements compared to traditional on-premise data warehouse solutions. First, software is deployed on elastic resources. Consequently, systems should support elasticity in order to match workload requirements and be cost-effective. We present an elastic scaling mechanism for distributed database engines, combined with a partition manager that provides load balancing while minimizing partition reassignments in the case of elastic scaling. Furthermore we introduce a buffer pre-heating strategy that allows to mitigate a cold start after scaling and leads to an immediate performance benefit using scaling. Second, cloud based systems are accessible and available from nearly everywhere. Consequently, data is frequently ingested from numerous endpoints, which differs from bulk loads or ETL pipelines in a traditional data warehouse solution. Many users do not define database constraints in order to avoid transaction aborts due to conflicts or to speed up data ingestion. To mitigate this issue we introduce the concept of PatchIndexes, which allow the definition of approximate constraints. PatchIndexes maintain exceptions to constraints, make them usable in query optimization and execution and offer efficient update support. The concept can be applied to arbitrary constraints and we provide examples of approximate uniqueness and approximate sorting constraints. Moreover, we show how PatchIndexes can be exploited to define advanced constraints like an approximate multi-key partitioning, which offers robust query performance over workloads with different partition key requirements. Third, data-centric workloads changed over the last decade. Besides traditional SQL workloads for business intelligence, data science workloads are of significant importance nowadays. For these cases the database system might only act as data delivery, while the computational effort takes place in data science or machine learning (ML) environments. As this workflow has several drawbacks, we follow the goal of pushing advanced analytics towards the database engine and introduce the Grizzly framework as a DataFrame-to-SQL transpiler. Based on this we identify user-defined functions (UDFs) and machine learning inference as important tasks that would benefit from a deeper engine integration and investigate approaches to push these operations towards the database engine

    Logarithmic delocalization of random Lipschitz functions on honeycomb and other lattices

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    We study random one-Lipschitz integer functions ff on the vertices of a finite connected graph, sampled according to the weight W(f)=v,wEcI{f(v)=f(w)}W(f) = \prod_{\langle v, w \rangle \in E} \mathbf{c}^{ \mathbb{I} \{ f(v) = f(w) \} } where c1\mathbf{c} \geq 1, and restricted by a boundary condition. For planar graphs, this is arguably the simplest ``2D random walk model'', and proving the convergence of such models to the Gaussian free field (GFF) is a major open question. Our main result is that for subgraphs of the honeycomb lattice (and some other cubic planar lattices), with flat boundary conditions and 1c21 \leq \mathbf{ c } \leq 2, such functions exhibit logarithmic variations. This is in line with the GFF prediction and improves a non-quantitative delocalization result by P. Lammers. The proof goes via level-set percolation arguments, including a renormalization inequality and a dichotomy theorem for level-set loops. In another direction, we show that random Lipschitz functions have bounded variance whenever the wired FK-Ising model with p=11/cp=1-1/\mathbf{c} percolates on the same lattice (corresponding to c>2+3\mathbf{c} > 2 + \sqrt{3} on the honeycomb lattice). Via a simple coupling, this also implies, perhaps surprisingly, that random homomorphisms are localized on the rhombille lattice.Comment: 70 pages; 10 figures with 20 illustration

    The grid-minor theorem revisited

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    We prove that for every planar graph XX of treedepth hh, there exists a positive integer cc such that for every XX-minor-free graph GG, there exists a graph HH of treewidth at most f(h)f(h) such that GG is isomorphic to a subgraph of HKcH\boxtimes K_c. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor
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