507 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On stability of metric spaces and Kalton's property
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 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
We introduce an annotated extension of treewidth that measures the
contribution of a vertex set to the treewidth of a graph This notion
provides a graph distance measure to some graph property : A
vertex set is a -treewidth modulator of to if the
treewidth of in is at most and its removal gives a graph in
This notion allows for a version of the Graph Minors Structure
Theorem (GMST) that has no need for apices and vortices: -minor free
graphs are those that admit tree-decompositions whose torsos have
-treewidth modulators to some surface of Euler-genus 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 is some particular surface and define a ``surface
extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum
for which admits a tree-decomposition whose torsos have a -treewidth
modulator to being embeddable in We identify a finite collection
of parametric graphs and prove that the minor-exclusion
of the graphs in precisely determines the asymptotic
behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface It
follows that the collection bijectively corresponds to
the ``surface obstructions'' for i.e., surfaces that are minimally
non-contained in $\Sigma.
Planar Disjoint Paths, Treewidth, and Kernels
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . 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 unless coNP
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 , where 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 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 , 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
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 -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 , there exists an integer such that every graph with no
subdivision of is (properly) conflict-free -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 is (properly) conflict-free -choosable.
This result applies to -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
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 .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . 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 -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 -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Modern data analytics in the cloud era
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
We study random one-Lipschitz integer functions on the vertices of a
finite connected graph, sampled according to the weight where
, 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 , 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
percolates on the same lattice (corresponding to 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
We prove that for every planar graph of treedepth , there exists a
positive integer such that for every -minor-free graph , there exists
a graph of treewidth at most such that is isomorphic to a
subgraph of . 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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