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Quantum and approximation algorithms for Maximum Witnesses of Boolean Matrix Products 7th

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Quantum and Approximation Algorithms for Maximum Witnesses o...

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

作者： Kowaluk, Miroslaw Lingas, Andrzej Institute of Informatics University of Warsaw Warsaw Poland Department of Computer Science Lund University Lund22100 Sweden

ISBN: (纸本)9783030678982

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n× n Boolean matrices of the form O(n2.575) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time O~ (n2+λ/2) = O~ (n2.434), where λ satisfies the equation ω(1,λ,1)=1+1.5λ and ω(1, λ, 1 ) is the exponent of the multiplication of an n× nλ matrix by an nλ× n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most in time O~((n/)nω(1,logn,1)). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i, j] of the product matrix C a witness of rank O(⌈ WC(i, j) / k⌉ ), where WC(i, j) is the number of witnesses for C[i, j], with high probability. The algorithm runs in O~ (nωk0.4653+ n2+o(1)k) time, where ω= ω(1, 1, 1 ). © 2021, Springer Nature Switzerland AG.

关键词： approximation algorithms

Certified approximation algorithms for the fermat point and n-ellipses 29

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Certified approximation algorithms for the fermat point and ...

29th Annual European Symposium on algorithms, ESA 2021

作者： Junginger, Kolja Mantas, Ioannis Papadopoulou, Evanthia Suderland, Martin Yap, Chee Faculty of Informatics Università della Svizzera Italiana Lugano Switzerland Courant Institute New York University NY United States

ISBN: (纸本)9783959772044

Given a set A of n points in Rd with weight function w : A → R>0, the Fermat distance function is φ(x):= Pa∈A w(a)x−a. A classic problem in facility location dating back to 1643, is to find the Fermat point x∗, the point that minimizes the function φ. We consider the problem of computing a point xe∗ that is an Ε-approximation of x∗ in the sense that xe∗ − x∗ ∗). We devise a certified subdivision algorithm for computing xe∗, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x∗, turning it into an Ε-approximate Fermat point algorithm. Our second problem is the certified construction of Ε-isotopic approximations of n-ellipses. These are the level sets φ−1(r) for r > φ(x∗) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques. © Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap;licensed under Creative Commons License CC-BY 4.0

关键词： approximation algorithms

approximation algorithms FOR SIZE-CONSTRAINED NON-MONOTONE SUBMODULAR MAXIMIZATION IN DETERMINISTIC LINEAR TIME

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arXiv 2021年

作者： Chen, Yixin Kuhnle, Alan Department of Computer Science & Engineering Texas A&M University College StationTX United States

In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and nonlinear regression. We provide the first deterministic, linear-time approximation algorithms for this problem that do not assume the objective is monotone. We present three deterministic, linear-time algorithms: a single-pass streaming algorithm with a ratio of 23.313 + ϵ, which is the first linear-time streaming algorithm;a simpler deterministic linear-time algorithm with a ratio of 11.657;and a (4 + O(ϵ))-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of e + ϵ in Oϵ(n log(n)) time, close to the best known expected ratio of e − 0.121 in polynomial time. Copyright © 2021, The Authors. All rights reserved.

关键词： approximation algorithms

FPT and FPT-approximation algorithms for unsplittable flow on trees 29

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FPT and FPT-approximation algorithms for unsplittable flow o...

29th Annual European Symposium on algorithms, ESA 2021

作者： Martínez-Muñoz, Tomás Wiese, Andreas Mathematical Engineering Department University of Chile Santiago Chile Department of Industrial Engineering University of Chile Santiago Chile

ISBN: (纸本)9783959772044

We study the unsplittable flow on trees (UFT) problem in which we are given a tree with capacities on its edges and a set of tasks, where each task is described by a path and a demand. Our goal is to select a subset of the given tasks of maximum size such that the demands of the selected tasks respect the edge capacities. The problem models throughput maximization in tree networks. The best known approximation ratio for (unweighted) UFT is O(log n). We study the problem under the angle of FPT and FPT-approximation algorithms. We prove that UFT is FPT if the parameters are the cardinality k of the desired solution and the number of different task demands in the input, UFT is FPT under (1 + δ)-resource augmentation of the edge capacities for parameters k and 1/δ, and UFT admits an FPT-5-approximation algorithm for parameter k. One key to our results is to compute structured hitting sets of the input edges which partition the given tree into O(k) clean components. This allows us to guess important properties of the optimal solution. Also, in some settings we can compute core sets of subsets of tasks out of which at least one task i is contained in the optimal solution. These sets have bounded size, and hence we can guess this task i easily. A consequence of our results is that the integral multicommodity flow problem on trees is FPT if the parameter is the desired amount of sent flow. Also, even under (1 + δ)-resource augmentation UFT is APX-hard, and hence our FPT-approximation algorithm for this setting breaks this boundary. © Tomás Martínez-Muñoz and Andreas Wiese;licensed under Creative Commons License CC-BY 4.0

关键词： approximation algorithms

Better approximation algorithms for Maximum Weight Internal Spanning Trees in Cubic Graphs and Claw-Free Graphs 15th

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Better Approximation Algorithms for Maximum Weight Internal ...

15th International Conference on algorithms and Computation, WALCOM 2021

作者： Biniaz, Ahmad School of Computer Science University of Windsor Windsor Canada

ISBN: (纸本)9783030682101

Given a connected vertex-weighted graph G, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of G that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor 1/2 (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor 7/12. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least 34-3n times the total weight of all vertices, where n is the number of vertices of G. This ratio is almost tight for large values of n. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least 35-1n times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors 34-ϵ and 35-ϵ for the MaxwIST problem in cubic graphs and claw-free graphs having no degree-2 vertices, for any ϵ> 0. The new algorithms are short (compared to that of Chen et al.) and fairly simple as they employ a variant of the depth-first search algorithm. Moreover, they take linear time while previous algorithms for similar problem instances are super-linear. © 2021, Springer Nature Switzerland AG.

关键词： approximation algorithms

approximation algorithms for LCS and LIS with truly improved running times

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arXiv 2021年

作者： Rubinstein, Aviad Seddighin, Saeed Song, Zhao Sun, Xiaorui Stanford University Toyota Technological Institute Chicago United States Adobe Research University of Illinois

Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence problem (LIS) which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size over the input size. We denote this ratio by λ and obtain the following results for LCS and LIS without any prior knowledge of λ. • A truly subquadratic time algorithm for LCS with approximation factor Ω(λ3). • A truly sublinear time algorithm for LIS with approximation factor Ω(λ3). Triangle inequality was recently used by Boroujeni, Ehsani, Ghodsi, HajiAghayi and Seddighin [BEG+18] and Charkraborty, Das, Goldenberg, Koucky and Saks [CDG+18] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of triangle inequality to non-metric settings. © 2021, CC BY-NC-SA.

关键词： approximation algorithms

Improved approximation algorithms for 2-dimensional knapsack: Packing into multiple L-shapes, spirals, and more 37

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Improved approximation algorithms for 2-dimensional knapsack...

37th International Symposium on Computational Geometry, SoCG 2021

作者： Gálvez, Waldo Grandoni, Fabrizio Khan, Arindam Ramírez-Romero, Diego Wiese, Andreas Department of Computer Science TU München Germany IDSIA USI-SUPSI Lugano Switzerland Department of Computer Science and Automation Indian Institute of Science Bangalore India Department of Mathematical Engineering Universidad de Chile Santiago Chile Department of Industrial Engineering Center for Mathematical Modeling Universidad de Chile Santiago Chile

ISBN: (纸本)9783959771849

In the 2-Dimensional Knapsack problem (2DK) we are given a square knapsack and a collection of n rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is 17/9 + Ε © Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese;licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

关键词： approximation algorithms

approximation algorithms for 1-Wasserstein distance between persistence diagrams

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arXiv 2021年

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

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. 2012 ACM Subject Classification Theory of computation approximation algorithms analysis. © 2021, CC BY.

关键词： approximation algorithms

approximation algorithms for the random-field Ising model

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arXiv 2021年

作者： Helmuth, Tyler Lee, Holden Perkins, Will Ravichandran, Mohan Wu, Qiang Durham University Mathematical Sciences Department Duke University Department of Mathematics University of Illinois at Chicago Department of Mathematics Statistics and Computer Science Bogazici University Department of Mathematics University of Illinois at Urbana-Champaign Department of Mathematics

Approximating the partition function of the ferromagnetic Ising model with general external fields is known to be #BIS-hard in the worst case, even for bounded-degree graphs, and it is widely believed that no polynomial-time approximation scheme exists. This motivates an average-case question: are there classes of instances for which polynomial-time approximation schemes exist? We investigate this question for the random field Ising model on graphs with maximum degree ∆. We establish the existence of fully polynomial-time approximation schemes and samplers with high probability over the random fields if the external fields are IID Gaussians with variance larger than a constant depending only on the inverse temperature and ∆. The main challenge comes from the positive density of vertices at which the external field is small. These regions, which may have connected components of size Θ(log n), are a barrier to algorithms based on establishing a zero-free region, and cause worst-case analyses of Glauber dynamics to fail. The analysis of our algorithm is based on percolation on a self-avoiding walk tree. Copyright © 2021, The Authors. All rights reserved.

关键词： approximation algorithms