791 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

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    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on nn-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth dd and using kk labels, we can solve - Independent Set in time 2O(dk)⋅nO(1)2^{O(dk)}\cdot n^{O(1)} using O(dk2log⁡n)O(dk^2\log n) space; - Max Cut in time nO(dk)n^{O(dk)} using O(dklog⁡n)O(dk\log n) space; and - Dominating Set in time 2O(dk)⋅nO(1)2^{O(dk)}\cdot n^{O(1)} using nO(1)n^{O(1)} space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with dd if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

    Improved Approximation Algorithms for Steiner Connectivity Augmentation Problems

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    The Weighted Connectivity Augmentation Problem is the problem of augmenting the edge-connectivity of a given graph by adding links of minimum total cost. This work focuses on connectivity augmentation problems in the Steiner setting, where we are not interested in the connectivity between all nodes of the graph, but only the connectivity between a specified subset of terminals. We consider two related settings. In the Steiner Augmentation of a Graph problem (kk-SAG), we are given a kk-edge-connected subgraph HH of a graph GG. The goal is to augment HH by including links and nodes from GG of minimum cost so that the edge-connectivity between nodes of HH increases by 1. In the Steiner Connectivity Augmentation Problem (kk-SCAP), we are given a Steiner kk-edge-connected graph connecting terminals RR, and we seek to add links of minimum cost to create a Steiner (k+1)(k+1)-edge-connected graph for RR. Note that kk-SAG is a special case of kk-SCAP. All of the above problems can be approximated to within a factor of 2 using e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this work, we leverage the framework of Traub and Zenklusen to give a (1+ln⁥2+Δ)(1 + \ln{2} +\varepsilon)-approximation for the Steiner Ring Augmentation Problem (SRAP): given a cycle H=(V(H),E)H = (V(H),E) embedded in a larger graph G=(V,EâˆȘL)G = (V, E \cup L) and a subset of terminals R⊆V(H)R \subseteq V(H), choose a subset of links S⊆LS \subseteq L of minimum cost so that (V,EâˆȘS)(V, E \cup S) has 3 pairwise edge-disjoint paths between every pair of terminals. We show this yields a polynomial time algorithm with approximation ratio (1+ln⁥2+Δ)(1 + \ln{2} + \varepsilon) for 22-SCAP. We obtain an improved approximation guarantee of (1.5+Δ)(1.5+\varepsilon) for SRAP in the case that R=V(H)R = V(H), which yields a (1.5+Δ)(1.5+\varepsilon)-approximation for kk-SAG for any kk

    An Approximation Algorithm for Two-Edge-Connected Subgraph Problem via Triangle-free Two-Edge-Cover

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    The 22-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. In the problem, we are given an undirected graph GG, and the objective is to find a 22-edge-connected spanning subgraph HH of GG with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with factor 1.3261.326, which was the best approximation ratio. In this paper, we give a (1.3+Δ)(1.3+\varepsilon)-approximation algorithm for 2-ECSS, where Δ\varepsilon is an arbitrary positive fixed constant, which improves the previously known best approximation ratio. In our algorithm, we compute a minimum triangle-free 22-edge-cover in GG with the aid of the algorithm for finding a maximum triangle-free 22-matching given by Hartvigsen. Then, with the obtained triangle-free 22-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli

    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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    Realizability of Free Spaces of Curves

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    The free space diagram is a popular tool to compute the well-known Fr\'echet distance. As the Fr\'echet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often, the question arises whether a certain pattern in the free space diagram is "realizable", i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore, we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in R>2\mathbb{R}^{> 2}, showing ∃R\exists\mathbb{R}-hardness. We use this to show that for curves in R≄2\mathbb{R}^{\ge 2}, the realizability problem is ∃R\exists\mathbb{R}-complete, both for continuous and for discrete Fr\'echet distance. We prove that the continuous case in R1\mathbb{R}^1 is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in R1\mathbb{R}^1, we show that the problem becomes solvable in polynomial time.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms And Computations (ISAAC 2023

    Demand Response in Smart Grids

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    The Special Issue “Demand Response in Smart Grids” includes 11 papers on a variety of topics. The success of this Special Issue demonstrates the relevance of demand response programs and events in the operation of power and energy systems at both the distribution level and at the wide power system level. This reprint addresses the design, implementation, and operation of demand response programs, with focus on methods and techniques to achieve an optimized operation as well as on the electricity consumer

    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 ĂŒber feinkörnige Algorithmen und KomplexitĂ€t untersuchen wir ausgabesensitive Algorithmen und Algorithmen mit sublinearer Lauf-zeit fĂŒr zwei wichtige Probleme. (1) DĂŒnne Faltungen: Die Berechnung der Faltung zweier Vektoren ist ein grundlegendes algorithmisches Primitiv, das in allen Bereichen der Informatik und des Ingenieurwesens Anwendung findet. FĂŒr das dĂŒnne Faltungsproblem nehmen wir an, dass die Eingabe- und Ausgabevektoren höchstens t EintrĂ€ge ungleich Null haben, und das Ziel ist, Algorithmen mit Laufzeiten in AbhĂ€ngigkeit von t zu entwickeln. FĂŒr den speziellen Fall, dass alle EintrĂ€ge nicht-negativ sind, was insbesondere fĂŒr den Entwurf von Algorithmen relevant ist, ist seit zwanzig Jahren bekannt, dass dĂŒnn 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 fĂŒr dĂŒnne nichtnegative Faltungen. Wir stellen auch eine Anwendung dieser Ergebnisse vor, die zu scheinbar unverwandten feinkörnigen unteren Schranken gegen Distanzorakel in Graphen fĂŒhrt. (2) Sublineare Editierdistanz: Die Editierdistanz zweier Zeichenketten ist ein gut untersuchtes Ähnlichkeitsmaß mit zahlreichen Anwendungen in der Computerbiologie. WĂ€hrend 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 gefĂŒhrt. 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
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