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SSRN 2023年

作者： Jana, Saswata Mandal, Partha Sarathi Department of Mathematics Indian Institute of Technology Guwahati Assam Guwahati781039 India

The coordination among drones and ground vehicles for last-mile delivery has gained significant interest in recent years. In this paper, we study \textit{multiple drone delivery scheduling problem} (MDSP) \cite{Betti_ICDCN22} for last-mile delivery, where we have a set of drones with an identical battery budget and a set of delivery locations;along with profit for delivery, cost, and delivery time intervals. The objective of the MDSP is to find conflict-free schedules for each drone such that the total profit gained is maximum subject to the battery constraint of the drones. This paper proposes an optimal pseudo-polynomial approximation algorithm and a fully polynomial time approximation scheme (FPTAS) for the \textit{single drone delivery scheduling problem} (SDSP). Also, we propose two approximation algorithms of factor $\fracapproximationapproximation$ for the MDSP with some constraints on the number of drones. We also formulate the problem for the heterogeneous case, where the drones have non-identical battery capacities, and propose a $\fracapproximation{6}\rho$-approximation algorithm, where $\rho$ is the ratio between the minimum and maximum battery capacity of the drones. © 2023, The Authors. All rights reserved.

关键词： approximation algorithms

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approximation algorithms for Norm Multiway Cut

arXiv

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arXiv 2023年

作者： Carlson, Charlie Jafarov, Jafar Makarychev, Konstantin Makarychev, Yury Shan, Liren University of Colorado Boulder United States Toyota Technological Institute Chicago United States Northwestern University United States

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced p-norm Multiway, a generalization of the problem, in which the goal is to minimize the p norm of the edge boundaries of k parts. We provide an O(log1/2 n log1/2+1/p k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log3/2 n log1/2 k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log1/2 n log7/2 k) approximation algorithm with a weaker oracle and an O(log1/2 n log5/2 k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n1/4−Ε approximation algorithm for every Ε > 0 assuming the Hypergraph Dense-vs-Random Conjecture. © 2023, CC BY-NC-SA.

关键词： approximation algorithms

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approximation algorithms for Steiner Connectivity Augmentation

arXiv

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arXiv 2023年

作者： Hathcock, Daniel Zlatin, Michael

We consider connectivity augmentation problems in the Steiner setting, where the goal is to augment the edge-connectivity between a specified subset of terminal nodes. In the Steiner Augmentation of a Graph problem (k-SAG), we are given a k-edge-connected subgraph H of a graph G. The goal is to augment H by including links from G of minimum cost so that the edge-connectivity between nodes of H increases by 1. This is a generalization of the Weighted Connectivity Augmentation Problem, in which only links between pairs of nodes in H are available for the augmentation. In the Steiner Connectivity Augmentation Problem (k-SCAP), we are given a Steiner k-edge-connected graph connecting terminals R, and we seek to add links of minimum cost to create a Steiner (k + 1)-edge-connected graph for R. Note that k-SAG is a special case of k-SCAP. The results of Ravi, Zhang and Zlatin for the Steiner Tree Augmentation problem yield a (1.5+Ε)-approximation for 1-SCAP and for k-SAG when k is odd [RZZ23]. In this work, we give a (1 + ln 2 + Ε)-approximation for the Steiner Ring Augmentation Problem (SRAP). This yields a polynomial time algorithm with approximation ratio (1 + ln 2 + Ε) for 2-SCAP. We obtain an improved approximation guarantee for SRAP when the ring consists of only terminals, yielding a (1.5 + Ε)-approximation for k-SAG for any k. Copyright © 2023, The Authors. All rights reserved.

关键词： approximation algorithms

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approximation algorithms for demand strip packing 24

Approximation algorithms for demand strip packing

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24th International Conference on approximation algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021

作者： Gálvez, Waldo Grandoni, Fabrizio Ameli, Afrouz Jabal Khodamoradi, Kamyar Technische Universität München Germany IDSIA USI-SUPSI Lugano Switzerland Universitä Würzburg Germany

ISBN: (纸本)9783959772075

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3 + Ε)approximation algorithm for any constant Ε > 0. We also achieve best-possible approximation factors for some relevant special cases. © Waldo Gálvez, Fabrizio Grandoni, Afrouz Jabal Ameli, and Kamyar Khodamoradi;licensed under Creative Commons License CC-BY 4.0

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approximation algorithms for the Balanced Optimization Splicing Problem in Undirected Graph 12th

Approximation Algorithms for the Balanced Optimization Splic...

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12th EAI International Conference on Simulation Tools and Techniques, SIMUtools 2020

作者： Wen, Yongsong School of Management Wuhan University of Technology Wuhan430070 China School of Mathematics and Statistics GuiZhou University of Finance and Economics Guiyang550025 China

ISBN: (纸本)9783030727918

One-dimensional bin packing problem and balanced optimization problem are two classical problems in combinatorial optimization, inspired by this, we research a balanced optimization splicing problem: given a weight connected G(V, E;w), a balanced spanning tree structure A and a specific material of fixed length , where w: E→ Q+ (or w: E→ Z+ ), each edge in the graph G is allowed to be greater than or equal to , we will use this specific material to splice a subgraph T from graph G with balanced spanning tree structure A, and these edges spliced in such required structures are supposed to be cut from some pieces of a specific material of fixed length , the objective is to minimize the amount of special material used when splicing all the edges of the subgraph T with the given material. In this paper, we consider three kinds of balanced spanning tree structures for this problem and design three approximation algorithms. © ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2021.

关键词： approximation algorithms

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approximation algorithms for the Euclidean Dispersion Problems 33

Approximation Algorithms for the Euclidean Dispersion Proble...

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33rd Canadian Conference on Computational Geometry, CCCG 2021

作者： Mishra, Pawan K. Das, Gautam K. Department of Computer Science and Engineering Indian Institute of Technology Guwahati India Department of Mathematics Indian Institute of Technology Guwahati India

In this article, we consider the Euclidean dispersion problems. Let P = {p1, p2, . . ., pn} be a set of n points in R2. For each point p ∈ P and S ⊆ P, we define costγ(p, S) as the sum of Euclidean distance from p to the nearest γ point in S \ {p}. We define costγ(S) = minp∈S{costγ(p, S)} for S ⊆ P. In the γdispersion problem, a set P of n points in R2 and a positive integer k ∈ [γ + 1, n] are given. The objective is to find a subset S ⊆ P of size k such that costγ(S) is maximized. We consider both 2-dispersion and 1-dispersion problem in R2. Along with these, we also consider 2-dispersion problem when points are placed on a line. In this paper, we propose a simple polynomial time (2√3 + ∊)-factor approximation algorithm for the 2-dispersion problem, for any ∊ > 0, which is an improvement over the best known approximation factor 4√3 [Amano, K. and Nakano, S. I., An approximation algorithm for the 2-dispersion problem, IEICE Transactions on Information and Systems, Vol. 103(3), pp. 506-508, 2020]. Next, we develop a common framework for designing an approximation algorithm for the Euclidean dispersion problem. With this common framework, we improve the approximation factor to 2√3 for the 2-dispersion problem in R2. Using the same framework, we propose a polynomial time algorithm, which returns an optimal solution for the 2-dispersion problem when points are placed on a line. Moreover, to show the effectiveness of the framework, we also propose a 2-factor approximation algorithm for the 1-dispersion problem in R2. © 2021 Meng He and Don Sheehy.

关键词： approximation algorithms

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approximation algorithms for 1-wasserstein distance between persistence diagrams 19

Approximation algorithms for 1-wasserstein distance between ...

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19th International Symposium on Experimental algorithms, SEA 2021

作者： Chen, Samantha Wang, Yusu University of California at San Diego San diegoCA United States

ISBN: (纸本)9783959771856

Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github. © Samantha Chen and Yusu Wang;licensed under Creative Commons License CC-BY 4.0 19th International Symposium on Experimental algorithms (SEA 2021).

关键词： approximation algorithms

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approximation algorithms for Orthogonal Line Centers 7th

Approximation Algorithms for Orthogonal Line Centers

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7th International Conference on algorithms and Discrete Applied Mathematics, CALDAM 2021

作者： Das, Arun Kumar Das, Sandip Mukherjee, Joydeep Indian Statistical Institute Kolkata India Ramakrishna Mission Vivekananda Educational and Research Institute Howrah India

ISBN: (纸本)9783030678982

k orthogonal line center problem computes a set of k axis-parallel lines for a given set of points in 2D such that the maximum among the distance between each point to its nearest line is minimized. A 2-factor approximation algorithm and a (74,32) bi-criteria approximation algorithm is presented for the problem. Both of them are deterministic approximation algorithms, having sub-quadratic running time and not based on linear programming. © 2021, Springer Nature Switzerland AG.

关键词： approximation algorithms

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approximation algorithms for Connectivity Augmentation Problems 1

引用

16th International Computer Science Symposium in Russia, CSR 2021

作者： Nutov, Zeev The Open University of Israel Ra’anana Israel

ISBN: (数字)9783030794163

ISBN: (纸本)9783030794156

In Connectivity Augmentation problems we are given a graph H= (V, EH) and an edge set E on V, and seek a min-size edge set J⊆ E such that H∪ J has larger edge/node connectivity than H. In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by 1. In the Block-Tree Augmentation problem H is connected and H∪ S should be 2-connected. In Leaf-to-Leaf Connectivity Augmentation problems every edge in E connects minimal deficient sets. For this version we give a simple combinatorial approximation algorithm with ratio 5/3, improving the 1.91 approximation of [6] (see also [23]), that applies for the general case. We also show by a simple proof that if the Steiner Tree problem admits approximation ratio α then the general version admits approximation ratio 1 + ln (4 - x) + ϵ, where x is the solution to the equation 1 + ln (4 - x) = α+ (α- 1 ) x. For the currently best value of α= ln 4 + ϵ [7] this gives ratio 1.942. This is slightly worse than the ratio 1.91 of [6], but has the advantage of using Steiner Tree approximation as a "black box". In the Element Connectivity Augmentation problem we are given a graph G= (V, E), S⊆ V, and connectivity requirements r= { r(u, v): u, v∈ S}. The goal is to find a min-size set J of new edges on S such that for all u, v∈ S the graph G∪ J contains r(u, v) uv-paths such that no two of them have an edge or a node in V\ S in common. The problem is NP-hard even when rmax=maxu,v∈Sr(u,v)=2. We obtain ratio 3/2, improving the previous ratio 7/4 of [22]. For the case of degree bounds on S we obtain the same ratio with just + 1 degree violation, which is tight, since deciding whether there exists a feasible solution is NP-hard even when rmax= 2. © 2021, Springer Nature Switzerland AG.

关键词： approximation algorithms

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approximation algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

Approximation Algorithms for the Bottleneck Asymmetric Trave...

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作者： An, Hyung-Chan Kleinberg, Robert Shmoys, David B. Department of Computer Science Yonsei University 50 Seodaemun-gu Seoul03722 Korea Republic of Department of Computer Science Cornell University IthacaNY14853 United States School of ORIE Department of Computer Science Cornell University IthacaNY14853 United States

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n/ log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M ...dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem. © 2021 Association for Computing Machinery.

关键词： approximation algorithms

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