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How to Make Your approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable approximation algorithms of Functions with Low Sensitivity 26

How to Make Your Approximation Algorithm Private: A Black-Bo...

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26th International Conference on approximation algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023

作者： Blocki, Jeremiah Grigorescu, Elena Mukherjee, Tamalika Zhou, Samson Purdue University West LafayetteIN United States University of California BerkeleyCA United States Rice University HoustonTX United States

ISBN: (纸本)9783959772969

We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Specifically, our results focus on algorithms A that output an approximation to a function f of the form (1 − α)f(x) − κ ≤ A(x) ≤ (1 + α)f(x) + κ, where κ ∈ R≥0 denotes additive error and α ∈ [0, 1) denotes multiplicative error can be"tuned" to small-enough values while incurring only a polynomial blowup in the running time/space. We show that such algorithms can be made differentially private without sacrificing accuracy, as long as the function f has small "global sensitivity". We achieve these results by applying the "smooth sensitivity" framework developed by Nissim, Raskhodnikova, and Smith (STOC 2007). Our framework naturally applies to transform non-private FPRAS and FPTAS algorithms into Ε-differentially private approximation algorithms where the former case requires an additional postprocessing step. We apply our framework in the context of sublinear-time and sublinear-space algorithms, while preserving the nature of the algorithm in meaningful ranges of the parameters. Our results include the first (to the best of our knowledge) Ε-edge differentially-private sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a minimum spanning tree of a graph whose accuracy holds with high probability. In the area of streaming algorithms, our results include Ε-DP algorithms for estimating Lp-norms, distinct elements, and weighted minimum spanning tree for both insertion-only and turnstile streams. Our transformation also provides a private version of the smooth histogram framework, which is commonly used for converting streaming algorithms into sliding window variants, and achieves a multiplicative approximation to many problems, such as estimating Lp-norms, distinct elements, and the length of the longest increasing subsequence. © 2023 Schloss Dagstuhl- Leibniz-

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approximation algorithms for Fair Range Clustering

arXiv

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arXiv 2023年

作者： Hotegni, Sèdjro S. Mahabadi, Sepideh Vakilian, Ali African Institute for Mathematical Sciences-Rwanda Rwanda Microsoft Research-Redmond United States United States

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P, d) where each point belongs to one of the different demographics (i.e., (Equation presented)) and a set of intervals [α1, β1], · · ·, [α, β] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum p-clustering cost (i.e., (Equation presented)) such that for each group i ∈ , |C ∩ Pi| ∈ [αi, βi]. In particular, the fair range p-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range p-clustering for all values of p ∈ [1, ∞). Copyright © 2023, The Authors. All rights reserved.

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approximation algorithms for Directed Weighted Spanners

arXiv

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arXiv 2023年

作者： Grigorescu, Elena Kumar, Nithish Lin, Young-San Purdue University United States Melbourne Business School Australia

In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. A more restricted variant of this problem was shown to be O(2log1−Ε n)-hard to approximate under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). This general formulation captures many well-studied network connectivity problems, including spanners, distance preservers, and Steiner forests. We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting: • An Õ(n4/5+Ε)-approximation algorithm for the pairwise weighted spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n3/5+Ε)-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on algorithms, 2020). • An Õ(n1/2+Ε)-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives an Õ(n1/2)-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. In the online setting, the terminal vertex pairs arrive one at a time, in an online fashion, and edges are required to be added irrevocably to the solution in order to satisfy the distance constraints, while approximately minimizing the cost. Our results include the followin

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approximation algorithms for the Integrated Path and Bin Packing Problem

SSRN

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SSRN 2023年

作者： Li, Weidong Sun, Ruiqing School of Mathematics and Statistics Yunnan University Kunming650504 China

In this paper, we consider the integrated path and bin packing problem, which to use the minimum number of unit-size bins to packing the arcs on the path between two specific vertices in a given directed graph. We propose a 3/2-approximation algorithm and an asymptotic polynomial time approximation scheme. © 2023, The Authors. All rights reserved.

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approximation algorithms for Job Scheduling with Reconfigurable Resources

arXiv

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arXiv 2023年

作者： Bergé, Pierre Chaikovskaia, Mari Gayon, Jean-Philippe Quilliot, Alain Université Clermont-Auvergne CNRS Mines de Saint-Etienne Clermont-Auvergne-INP LIMOS Clermont-Ferrand63000 France IMT Atlantique LS2N UMR CNRS 6004 NantesF-44307 France

We consider here the Multi Bot problem for the scheduling and the resource parametrization of jobs related to the production or the transportation of different products inside a given time horizon. Those jobs must meet known in advance demands. The time horizon is divided into several discrete identical periods representing each the time needed to proceed a job. The objective is to find a parametrization and a schedule for the jobs in such a way they require as less resources as possible. Though this problem derived from the applicative context of reconfigurable robots, we focus here on fundamental issues. We show that the resulting strongly NP-hard Multi Bot problem may be handled in a greedy way with an approximation ratio of 4/3. © 2023, CC BY.

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approximation algorithms for Preference Aggregation Using CP-Nets

arXiv

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arXiv 2023年

作者： Ali, Abu Mohammad Hammad Yang, Boting Zilles, Sandra University of Regina Canada

This paper studies the design and analysis of approximation algorithms for aggregating preferences over combinatorial domains, represented using Conditional Preference Networks (CP-nets). Its focus is on aggregating preferences over so-called swaps, for which optimal solutions in general are already known to be of exponential size. We first analyze a trivial 2-approximation algorithm that simply outputs the best of the given input preferences, and establish a structural condition under which the approximation ratio of this algorithm is improved to 4/3. We then propose a polynomial-time approximation algorithm whose outputs are provably no worse than those of the trivial algorithm, but often substantially better. A family of problem instances is presented for which our improved algorithm produces optimal solutions, while, for any Ε, the trivial algorithm cannot attain a (2−Ε)-approximation. These results may lead to the first polynomial-time approximation algorithm that solves the CP-net aggregation problem for swaps with an approximation ratio substantially better than 2. © 2023, CC BY.

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approximation algorithms for k-median problems on complex networks: theory and practice

arXiv

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arXiv 2023年

作者： Pozo, Roldan National Institute of Standards & Technology

Finding the k-median in a network involves identifying a subset of k vertices that minimize the total distance to all other vertices in a graph. This problem has been extensively studied in computer science, graph theory, operations research, and numerous areas due to its significance in a wide range of applications. While known to be computationally challenging (NP-hard) several approximation algorithms have been proposed, most with high-order polynomial-time complexity. However, the graph topology of complex networks with heavy-tailed degree distributions present characteristics that can be exploited to yield custom-tailored algorithms. We compare eight algorithms specifically designed for complex networks and evaluate their performance based on accuracy and efficiency for problems of varying sizes and application areas. Rather than relying on a small number of problems, we conduct over 16,000 experiments covering a wide range of network sizes and k-median values. While individual results vary, a few methods provide consistently good results. We draw general conclusions about how algorithms perform in practice and provide general guidelines for solutions. © 2023, CC BY.

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Tight approximation algorithms for Ordered Covering 1

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18th International Symposium on algorithms and Data Structures, WADS 2023

作者： Batra, Jatin Das, Syamantak Jha, Agastya Vibhuti Tata Institute of Fundamental Research Mumbai India Indraprastha Institute of Information Technology Delhi India Ecole Polytechnique Fédérale de Lausanne Lausanne Switzerland

ISBN: (数字)9783031389061

ISBN: (纸本)9783031389054

The classical unweighted set cover problem aims to pick a minimum number of subsets from a given family of subsets whose union would cover the universe of elements. The vertex cover problem is an important special case where the subsets and elements are respectively vertices and edges of a given undirected graph. On the other hand, the min-sum versions of both problems offer a different perspective - the objective is to find a linear ordering of the covering entities so that the sum of cover times of the elements is minimized. In this paper, we study common generalizations of these two classical problems. In particular, we study the Top- -norm problem for set cover and the even more general ordered norm (a linear combination of Top- norms) for vertex cover. *** give a polynomial time randomized rounding based (2 + Ε) -approx. for ordered norm vertex *** show that a natural greedy algorithm gives O(log (n/ ) ) -approx. for Top- set cover and show a matching hardness up to constants. At a technical level, we exploit a certain known progress property of the set cover greedy algorithm for the Top- set cover problem. For the ordered norm vertex cover problem, we employ a major shift from the standard independent rounding techniques used for the min-sum vertex cover problem. We instead use a slightly weaker version of dependent rounding on a carefully constructed bipartite assignment instance. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023.

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approximation algorithms for the square min-sum bin packing problem

arXiv

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arXiv 2023年

作者： Saraiva, Rachel Vanucchi Schouery, Rafael C.S. Institute of Computing University of Campinas Av. Albert Einstein 1251 Campinas São Paulo13083-852 Brazil

In this work, we study the square min-sum bin packing problem (SMSBPP), where a list of square items has to be packed into indexed square bins of dimensions 1 × 1 with no overlap between the areas of the items. The bins are indexed and the cost of packing each item is equal to the index of the bin in which it is placed in. The objective is to minimize the total cost of packing all items, which is equivalent to minimizing the average cost of items. The problem has applications in minimizing the average time of logistic operations such as cutting stock and delivery of products. We prove that classic algorithms for two-dimensional bin packing that order items in non-increasing order of size, such as Next Fit Decreasing Height or Any Fit Decreasing Height heuristics, can have an arbitrarily bad performance for SMSBPP. We, then, present a 5322-approximation and a PTAS for the problem. © 2023, CC BY.

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approximation algorithms for Packing Cycles and Paths in Complete Graphs

arXiv

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arXiv 2023年

作者： Zhao, Jingyang Xiao, Mingyu University of Electronic Science and Technology of China China

Given an edge-weighted (metric/general) complete graph with n vertices, the maximum weight (metric/general) k-cycle/path packing problem is to find a set of nk vertex-disjoint k-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric k-cycle packing, we improve the previous approximation ratio from 3/5 to 7/10 for k = 5, and from 7/8 · (1 − 1/k)2 for k > 5 to (7/8 − 0.125/k)(1 − 1/k) for constant odd k > 5 and to (formula presented). For metric k-path packing, we improve the approximation ratio from 7/8 ·(formula presented) for even 10 ≥ k ≥ 6. For the case of k = 4, we improve the approximation ratio from 3/4 to 5/6 for metric 4-cycle packing, from 2/3 to 3/4 for general 4-cycle packing, and from 3/4 to 14/17 for metric 4-path packing. © 2023, CC BY.

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