374 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Analysing trajectory similarity and improving graph dilation

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    In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph

    Parameterized Graph Modification Beyond the Natural Parameter

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    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

    Algorithms for sparse convolution and sublinear edit distance

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    In this PhD thesis on fine-grained algorithm design and complexity, we investigate output-sensitive and sublinear-time algorithms for two important problems. (1) Sparse Convolution: Computing the convolution of two vectors is a basic algorithmic primitive with applications across all of Computer Science and Engineering. In the sparse convolution problem we assume that the input and output vectors have at most t nonzero entries, and the goal is to design algorithms with running times dependent on t. For the special case where all entries are nonnegative, which is particularly important for algorithm design, it is known since twenty years that sparse convolutions can be computed in near-linear randomized time O(t log^2 n). In this thesis we develop a randomized algorithm with running time O(t \log t) which is optimal (under some mild assumptions), and the first near-linear deterministic algorithm for sparse nonnegative convolution. We also present an application of these results, leading to seemingly unrelated fine-grained lower bounds against distance oracles in graphs. (2) Sublinear Edit Distance: The edit distance of two strings is a well-studied similarity measure with numerous applications in computational biology. While computing the edit distance exactly provably requires quadratic time, a long line of research has lead to a constant-factor approximation algorithm in almost-linear time. Perhaps surprisingly, it is also possible to approximate the edit distance k within a large factor O(k) in sublinear time O~(n/k + poly(k)). We drastically improve the approximation factor of the known sublinear algorithms from O(k) to k^{o(1)} while preserving the O(n/k + poly(k)) running time.In dieser Doktorarbeit über feinkörnige Algorithmen und Komplexität untersuchen wir ausgabesensitive Algorithmen und Algorithmen mit sublinearer Lauf-zeit für zwei wichtige Probleme. (1) Dünne Faltungen: Die Berechnung der Faltung zweier Vektoren ist ein grundlegendes algorithmisches Primitiv, das in allen Bereichen der Informatik und des Ingenieurwesens Anwendung findet. Für das dünne Faltungsproblem nehmen wir an, dass die Eingabe- und Ausgabevektoren höchstens t Einträge ungleich Null haben, und das Ziel ist, Algorithmen mit Laufzeiten in Abhängigkeit von t zu entwickeln. Für den speziellen Fall, dass alle Einträge nicht-negativ sind, was insbesondere für den Entwurf von Algorithmen relevant ist, ist seit zwanzig Jahren bekannt, dass dünn besetzte Faltungen in nahezu linearer randomisierter Zeit O(t \log^2 n) berechnet werden können. In dieser Arbeit entwickeln wir einen randomisierten Algorithmus mit Laufzeit O(t \log t), der (unter milden Annahmen) optimal ist, und den ersten nahezu linearen deterministischen Algorithmus für dünne nichtnegative Faltungen. Wir stellen auch eine Anwendung dieser Ergebnisse vor, die zu scheinbar unverwandten feinkörnigen unteren Schranken gegen Distanzorakel in Graphen führt. (2) Sublineare Editierdistanz: Die Editierdistanz zweier Zeichenketten ist ein gut untersuchtes Ähnlichkeitsmaß mit zahlreichen Anwendungen in der Computerbiologie. Während die exakte Berechnung der Editierdistanz nachweislich quadratische Zeit erfordert, hat eine lange Reihe von Forschungsarbeiten zu einem Approximationsalgorithmus mit konstantem Faktor in fast-linearer Zeit geführt. Überraschenderweise ist es auch möglich, die Editierdistanz k innerhalb eines großen Faktors O(k) in sublinearer Zeit O~(n/k + poly(k)) zu approximieren. Wir verbessern drastisch den Approximationsfaktor der bekannten sublinearen Algorithmen von O(k) auf k^{o(1)} unter Beibehaltung der O(n/k + poly(k))-Laufzeit

    Deep material networks for efficient scale-bridging in thermomechanical simulations of solids

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    We investigate deep material networks (DMN). We lay the mathematical foundation of DMNs and present a novel DMN formulation, which is characterized by a reduced number of degrees of freedom. We present a efficient solution technique for nonlinear DMNs to accelerate complex two-scale simulations with minimal computational effort. A new interpolation technique is presented enabling the consideration of fluctuating microstructure characteristics in macroscopic simulations

    Quantum Computing for Airline Planning and Operations

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    Classical algorithms and mathematical optimization techniques have beenused extensively by airlines to optimize their profit and ensure that regulationsare followed. In this thesis, we explore which role quantum algorithmscan have for airlines. Specifically, we have considered the two quantum optimizationalgorithms; the Quantum Approximate Optimization Algorithm(QAOA) and Quantum Annealing (QA). We present a heuristic that integratesthese quantum algorithms into the existing classical algorithm, whichis currently employed to solve airline planning problems in a state-of-the-artcommercial solver. We perform numerical simulations of QAOA circuits andfind that linear and quadratic algorithm depth in the input size can be requiredto obtain a one-shot success probability of 0.5. Unfortunately, we areunable to find performance guarantees. Finally, we perform experiments withD-wave’s newly released QA machine and find that it outperforms 2000Q formost instances

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Mending Partial Solutions with Few Changes

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    In this paper, we study the notion of mending: given a partial solution to a graph problem, how much effort is needed to take one step towards a proper solution? For example, if we have a partial coloring of a graph, how hard is it to properly color one more node? In prior work (SIROCCO 2022), this question was formalized and studied from the perspective of mending radius: if there is a hole that we need to patch, how far do we need to modify the solution? In this work, we investigate a complementary notion of mending volume: how many nodes need to be modified to patch a hole? We focus on the case of locally checkable labeling problems (LCLs) in trees, and show that already in this setting there are two infinite hierarchies of problems: for infinitely many values 0 < ? ? 1, there is an LCL problem with mending volume ?(n^?), and for infinitely many values k ? 1, there is an LCL problem with mending volume ?(log^k n). Hence the mendability of LCL problems on trees is a much more fine-grained question than what one would expect based on the mending radius alone

    Parameterized Graph Modification Beyond the Natural Parameter

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