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arXiv 2022年

作者： Kulik, Ariel Mnich, Matthias Shachnai, Hadas CISPA Helmholtz Center for Information Security Germany Hamburg University of Technology Institute for Algorithms and Complexity Hamburg Germany Computer Science Department Technion Haifa Israel

We study the d-dimensional Vector Bin Packing (dVBP) problem, a generalization of Bin Packing with central applications in resource allocation and scheduling. In dVBP, we are given a set of items, each of which is characterized by a d-dimensional volume vector;the objective is to partition the items into a minimum number of subsets (bins), such that the total volume of items in each subset is at most 1 in each dimension. Our main result is an asymptotic approximation algorithm for dVBP that yields a ratio of (1+ln d−χ(d)+Ε) for all d ∈ N and any Ε > 0;here, χ(d) is some strictly positive function. This improves upon the best known asymptotic ratio of (1 + ln d + Ε) due to Bansal, Caprara and Sviridenko (SICOMP 2010) for any d > 3. By slightly modifying our algorithm to include an initial matching phase and applying a tighter analysis we obtain an asymptotic approximation ratio of (4/3 + Ε) for the special case of d = 2, thus substantially improving the previous best ratio of (3/2 + Ε) due to Bansal, Eliáš and Khan (SODA 2016). Our algorithm iteratively solves a configuration LP relaxation for the residual instance (from previous iterations) and samples a small number of configurations based on the solution for the configuration LP. While iterative rounding was already used by Karmarkar and Karp (FOCS 1982) to establish their celebrated result for classic (one-dimensional) Bin Packing, iterative randomized rounding is used here for the first time in the context of (Vector) Bin Packing. Our results show that iterative randomized rounding is a powerful tool for approximating dVBP, leading to simple algorithms with improved approximation guarantees. Copyright © 2022, The Authors. All rights reserved.

关键词： Vectors

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Balanced Independent and Dominating Sets on Colored Interval Graphs

arXiv

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arXiv 2020年

作者： Bhore, Sujoy Haunert, Jan-Henrik Klute, Fabian Li, Guangping Nöllenburg, Martin Indian Institute of Science Education and Research Bhopal India Geoinformation Group University of Bonn Bonn Germany Algorithms and Complexity Group TU Wien Vienna Austria

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely f-Balanced Independent Set (f-BIS) and f-Balanced Dominating Set (f-BDS). Let G = (V, E) be a vertex-colored interval graph with a k-coloring γ : V → {1,..., k} for some k ∈ N. A subset of vertices S ⊆ V is called f-balanced if S contains f vertices from each color class. In the f-BIS and f-BDS problems, the objective is to compute an independent set or a dominating set that is f-balanced. We show that both problems are NP-complete even on proper interval graphs. For the f-BIS problem we design two FPT algorithms, one parameterized by (f, k) for interval graphs and the other parameterized by the vertex cover number for general graphs. Moreover, for the optimization variant of 1-BIS on interval graphs, we show that a simple greedy approach achieves approximation ratio 2. Copyright © 2020, The Authors. All rights reserved.

关键词： Graphic methods

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Counting Vanishing Matrix-Vector Products

arXiv

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arXiv 2023年

作者： Brand, Cornelius Korchemna, Viktoriia Skotnica, Michael Simonov, Kirill Algorithms and Complexity Group TU Wien Austria Department of Applied Mathematics Charles University Prague Czech Republic Hasso Plattner Institute University of Potsdam Germany

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let v ∈ Qd be a rational vector, (T1, T2 . . . Tm) a list of d × d rational matrices, S ∈ Qh×d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices Ti1 , Ti2 , . . ., Tik from the list such that STik · · · Ti1 v = 0 ∈ Qh. In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d > 3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA’14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized the matrix dimensions and the order of the field. Copyright © 2023, The Authors. All rights reserved.

关键词： Matrix algebra

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Simulation aided, knowledge based routing for AGVs in a distribution warehouse 11

Simulation aided, knowledge based routing for AGVs in a dist...

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Simulation Winter Conference

作者： Alexander Klaas Christoph Laroque Wilhelm Dangelmaier Matthias Fischer Business Computing especially CIM Heinz Nixdorf Institute University of Paderborn Paderborn Germany Algorithms and Complexity Heinz Nixdorf Institute and Department of Computer Science University of Paderborn Paderborn Germany

Traditional routing algorithms for real world AGV systems in warehouses compute static paths, which can only be adjusted to a limited degree in the event of unplanned disturbances. In our approach, we aim for a higher reactivity in such events and plan small steps of a path incrementally. The current traffic situation and also up to date time constraints for each AGV can then be considered. We compute each step in real time based on empirical data stored in a knowledge base. It contains information covering a broad temporal horizon of the system to prevent costly decisions that may occur when only considering short term consequences. The knowledge is gathered through machine learning from the results of multiple experiments in a discrete event simulation during preprocessing. We implemented and experimentally evaluated the algorithm in a test scenario and achieve a natural robustness against delays and failures.

关键词： Vehicles Routing Heuristic algorithms Vehicle dynamics Knowledge based systems Computational modeling Planning

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Algorithmic Extensions of Dirac's Theorem

arXiv

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arXiv 2020年

作者： Fomin, Fedor V. Golovach, Petr A. Sagunov, Danil Simonov, Kirill Department of Informatics University of Bergen Norway St. Petersburg Department V.A. Steklov Institute of Mathematics Russia Algorithms and Complexity Group TU Wien Austria

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every n-vertex 2-connected graph G with minimum vertex degree δ ≥ 2 contains a cycle with at least min{2δ, n} vertices. In particular, if δ ≥ n/2, then G is Hamiltonian. The proof of Dirac's theorem is constructive, and it yields an algorithm computing the corresponding cycle in polynomial time. The combinatorial bound of Dirac's theorem is tight in the following sense. There are 2-connected graphs that do not contain cycles of length more than 2δ + 1. Also, there are non-Hamiltonian graphs with all vertices but one of degree at least n/2. This prompts naturally to the following algorithmic questions. For k ≥ 1, (A) How difficult is to decide whether a 2-connected graph contains a cycle of length at least min{2δ + k, n}? (B) How difficult is to decide whether a graph G is Hamiltonian, when at least n - k vertices of G are of degrees at least n/2 - k? The first question was asked by Fomin, Golovach, Lokshtanov, Panolan, Saurabh, and Zehavi. The second question is due to Jansen, Kozma, and Nederlof. Even for a very special case of k = 1, the existence of a polynomial-time algorithm deciding whether G contains a cycle of length at least min{2δ + 1, n} was open. We resolve both questions by proving the following algorithmic generalization of Dirac's theorem: If all but k vertices of a 2-connected graph G are of degree at least δ, then deciding whether G has a cycle of length at least min{2δ + k, n} can be done in time 2O(k) nO(1). The proof of the algorithmic generalization of Dirac's theorem builds on new graph-theoretical results that are interesting on their own. Copyright © 2020, The Authors. All rights reserved.

关键词： Hamiltonians

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Resource and service discovery for large-scale robot networks in disaster scenarios

Resource and service discovery for large-scale robot network...

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IEEE International Workshop on Safety, Security, and Rescue Robotics (SSRR)

作者： Jia Lei Du S. Ruhrup U. Witkowski U. Ruckert System and Circuit Technology Research Group Heinz Nixdorf Institute University of Paderborn Paderborn Germany Algorithms and Complexity Research Group Heinz Nixdorf Institute University of Paderborn Paderborn Germany

If robots are deployed in large numbers in disaster scenarios, the ability to discover and exchange resources and services with other robots in an open, heterogeneous, large-scale network would be essential for a successful operation. In this paper we present a discovery protocol that enables robots to efficiently locate resources and services available in large-scale networks. It is specifically designed for robot networks which are characterized by potentially highly dynamic network topologies and high service announcement-to-lookup ratios. The protocol exploits the position data of the robots to increase scalability and efficiency. A cell-based overlay structure is created, with master nodes in each cell. Through proactive intra-cell communication and reactive inter-cell communication, scalability is ensured and the effects of node movements on the overall network are minimized.

关键词： Large-scale systems Intelligent networks Robot sensing systems Mobile communication Mobile robots Scalability Robotics and automation Protocols Humans Robot vision systems

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Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time 40

Fully Dynamic Maximum Independent Sets of Disks in Polylogar...

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40th International Symposium on Computational Geometry, SoCG 2024

作者： Bhore, Sujoy Nöllenburg, Martin Tóth, Csaba D. Wulms, Jules Department of Computer Science and Engineering Indian Institute of Technology Bombay Mumbai India Institute of Logic and Computation Algorithms and Complexity Group TU Wien Austria Department of Mathematics California State University Northridge Los AngelesCA United States Department of Computer Science Tufts University MedfordMA United States Department of Mathematics and Computer Science TU Eindhoven Netherlands

ISBN: (纸本)9783959773164

A fundamental question is whether one can maintain a maximum independent set (MIS) in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. For a set of intervals, it is known that no dynamic algorithm can maintain an exact MIS in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate MIS in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n unit disks in the plane, we show that a 12-approximate MIS can be maintained with worst-case update time O(log n), and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension d, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain O(1 + Ε)-approximate MIS in truly sublinear update time, under standard complexity assumptions. Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (ESA 2021) that can smoothly transition from one independent set to another;hence it suffices to maintain a family of independent sets where the largest one is an O(1)-approximate MIS. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (DCG 2020) and Liu (SICOMP 2022), which generalizes the dynamic convex hull data structure by Chan (JACM 2010), and quickly yields a "replacement" disk (if any) when a disk in one of our independent sets is deleted. © Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms.

关键词： Data structures

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Dual-mode greedy algorithms can save energy 30

Dual-mode greedy algorithms can save energy

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30th International Symposium on algorithms and Computation, ISAAC 2019

作者： Geissmann, Barbara Leucci, Stefano Liu, Chih-Hung Penna, Paolo Proietti, Guido Department of Computer Science ETH Zürich Switzerland Department of Algorithms and Complexity Max Planck Institute for Informatics Germany Dipartimento di Ingegneria e Scienze Dell'Informazione e Matematica Università Dell'Aquila Italy Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" CNR Roma Italy

ISBN: (纸本)9783959771306

In real world applications, important resources like energy are saved by deliberately using so-called low-cost operations that are less reliable. Some of these approaches are based on a dual mode technology where it is possible to choose between high-energy operations (always correct) and low-energy operations (prone to errors), and thus enable to trade energy for correctness. In this work we initiate the study of algorithms for solving optimization problems that in their computation are allowed to choose between two types of operations: high-energy comparisons (always correct but expensive) and low-energy comparisons (cheaper but prone to errors). For the errors in low-energy comparisons, we assume the persistent setting, which usually makes it impossible to achieve optimal solutions without high-energy comparisons. We propose to study a natural complexity measure which accounts for the number of operations of either type separately. We provide a new family of algorithms which, for a fairly large class of maximization problems, return a constant approximation using only polylogarithmic many high-energy comparisons and only O(n log n) low-energy comparisons. This result applies to the class of p-extendible systems [24], which includes several NP-hard problems and matroids as a special case (p = 1). These algorithmic solutions relate to some fundamental aspects studied earlier in different contexts: (i) the approximation guarantee when only ordinal information is available to the algorithm;(ii) the fact that even such ordinal information may be erroneous because of low-energy comparisons and (iii) the ability to approximately sort a sequence of elements when comparisons are subject to persistent errors. Finally, our main result is quite general and can be parametrized and adapted to other error models. © Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, Paolo Penna, and Guido Proietti;licensed under Creative Commons License CC-BY

关键词： Approximation algorithms

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Parameterized complexity and approximability of directed odd cycle transversal

arXiv

引用

arXiv 2017年

作者： Lokshtanov, Daniel Ramanujan, M.S. Saurabh, Saket Zehavi, Meirav University of Bergen Bergen Norway Algorithms and Complexity Group TU Wien Vienna Austria Institute of Mathematical Sciences HBNI Chennai India University of Bergen Bergen Norway

A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time f(k)nO(1) unless FPT = W[1]. On the positive side, we give a factor 2 fixed parameter tractable (FPT) approximation algorithm for the problem. More precisely, our algorithm takes as input D and k, runs in time 2O(k2)nO(1), and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally, we provide evidence that there exists ϵ >0 such that DOCT does not admit a factor (1 + ϵ ) FPT-approximation algorithm. Copyright © 2017, The Authors. All rights reserved.

关键词： Directed graphs

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Minimum Link Fencing

arXiv

引用

arXiv 2022年

作者： Bhore, Sujoy Klute, Fabian Löffler, Maarten Nöllenburg, Martin Terziadis, Soeren Villedieu, Anaïs Department of Computer Science & Engineering Indian Institute of Technology Bombay India Department of Information and Computing Sciences Utrecht University Netherlands Algorithms and Complexity Group TU Wien Austria

We study a variant of the geometric multicut problem, where we are given a set P of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where P contains a polygon Q that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an O(n log n)-time algorithm for the case that the convex hull of P \ {Q} does not intersect Q. © 2022, CC BY.

关键词： Computational geometry

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