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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We consider the classic 1-center problem: Given a set P of n points in a metric space find the point in P that minimizes the maximum distance to the other points of P. We study the complexity of this problem in d-dimensional 𝓁_p-metrics and in edit and Ulam metrics over strings of length d. Our results for the 1-center problem may be classified based on d as follows.
- Small d. Assuming the hitting set conjecture (HSC), we show that when d = ω(log n), no subquadratic algorithm can solve the 1-center problem in any of the 𝓁_p-metrics, or in the edit or Ulam metrics.
- Large d. When d = Ω(n), we extend our conditional lower bound to rule out subquartic algorithms for the 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a (1+ε)-approximation for 1-center in the Ulam metric with running time O_{ε}̃(nd+n²√d).
We also strengthen some of the above lower bounds by allowing approximation algorithms or by reducing the dimension d, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of n strings each of length n, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

Amir Abboud, MohammadHossein Bateni, Vincent Cohen-Addad, Karthik C. S., and Saeed Seddighin. On Complexity of 1-Center in Various Metrics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abboud_et_al:LIPIcs.APPROX/RANDOM.2023.1, author = {Abboud, Amir and Bateni, MohammadHossein and Cohen-Addad, Vincent and Karthik C. S. and Seddighin, Saeed}, title = {{On Complexity of 1-Center in Various Metrics}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.1}, URN = {urn:nbn:de:0030-drops-188260}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.1}, annote = {Keywords: Center, Clustering, Edit metric, Ulam metric, Hamming metric, Fine-grained Complexity, Approximation} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Consider an agent exploring an unknown graph in search of some goal state. As it walks around the graph, it learns the nodes and their neighbors. The agent only knows where the goal state is when it reaches it. How do we reach this goal while moving only a small distance? This problem seems hopeless, even on trees of bounded degree, unless we give the agent some help. This setting with "help" often arises in exploring large search spaces (e.g., huge game trees) where we assume access to some score/quality function for each node, which we use to guide us towards the goal. In our case, we assume the help comes in the form of distance predictions: each node v provides a prediction f(v) of its distance to the goal vertex. Naturally if these predictions are correct, we can reach the goal along a shortest path. What if the predictions are unreliable and some of them are erroneous? Can we get an algorithm whose performance relates to the error of the predictions?
In this work, we consider the problem on trees and give deterministic algorithms whose total movement cost is only O(OPT + Δ ⋅ ERR), where OPT is the distance from the start to the goal vertex, Δ the maximum degree, and the ERR is the total number of vertices whose predictions are erroneous. We show this guarantee is optimal. We then consider a "planning" version of the problem where the graph and predictions are known at the beginning, so the agent can use this global information to devise a search strategy of low cost. For this planning version, we go beyond trees and give an algorithms which gets good performance on (weighted) graphs with bounded doubling dimension.

Siddhartha Banerjee, Vincent Cohen-Addad, Anupam Gupta, and Zhouzi Li. Graph Searching with Predictions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{banerjee_et_al:LIPIcs.ITCS.2023.12, author = {Banerjee, Siddhartha and Cohen-Addad, Vincent and Gupta, Anupam and Li, Zhouzi}, title = {{Graph Searching with Predictions}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {12:1--12:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.12}, URN = {urn:nbn:de:0030-drops-175158}, doi = {10.4230/LIPIcs.ITCS.2023.12}, annote = {Keywords: Algorithms with predictions, network algorithms, graph search} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds.
1) We give a polynomial-time approximation scheme (PTAS) for a diversified search ranking problem [Nikhil Bansal et al., 2010] whose objective is to minimizes the discounted cumulative gain. Our PTAS runs in time n^{2^O(log(1/ε)/ε)} ⋅ m^O(1) where n denotes the number of elements in the databases and m denotes the number of constraints. Complementing this result, we show that no PTAS can run in time f(ε) ⋅ (nm)^{2^o(1/ε)} assuming Gap-ETH and therefore our running time is nearly tight. Both our upper and lower bounds answer open questions from [Nikhil Bansal et al., 2010].
2) We next consider the Max-Sum Dispersion problem, whose objective is to select k out of n elements from a database that maximizes the dispersion, which is defined as the sum of the pairwise distances under a given metric. We give a quasipolynomial-time approximation scheme (QPTAS) for the problem which runs in time n^{O_ε(log n)}. This improves upon previously known polynomial-time algorithms with approximate ratios 0.5 [Refael Hassin et al., 1997; Allan Borodin et al., 2017]. Furthermore, we observe that reductions from previous work rule out approximation schemes that run in n^õ_ε(log n) time assuming ETH.
3) Finally, we consider a generalization of Max-Sum Dispersion called Max-Sum Diversification. In addition to the sum of pairwise distance, the objective also includes another function f. For monotone submodular function f, we give a quasipolynomial-time algorithm with approximation ratio arbitrarily close to (1-1/e). This improves upon the best polynomial-time algorithm which has approximation ratio 0.5 [Allan Borodin et al., 2017]. Furthermore, the (1-1/e) factor is also tight as achieving better-than-(1-1/e) approximation is NP-hard [Uriel Feige, 1998].

Amir Abboud, Vincent Cohen-Addad, Euiwoong Lee, and Pasin Manurangsi. Improved Approximation Algorithms and Lower Bounds for Search-Diversification Problems. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2022.7, author = {Abboud, Amir and Cohen-Addad, Vincent and Lee, Euiwoong and Manurangsi, Pasin}, title = {{Improved Approximation Algorithms and Lower Bounds for Search-Diversification Problems}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {7:1--7:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.7}, URN = {urn:nbn:de:0030-drops-163481}, doi = {10.4230/LIPIcs.ICALP.2022.7}, annote = {Keywords: Approximation Algorithms, Complexity, Data Mining, Diversification} }

Document

**Published in:** LIPIcs, Volume 192, 2nd Symposium on Foundations of Responsible Computing (FORC 2021)

Redistricting is the problem of dividing up a state into a given number k of regions (called districts) where the voters in each district are to elect a representative. The three primary criteria are: that each district be connected, that the populations of the districts be equal (or nearly equal), and that the districts are "compact". There are multiple competing definitions of compactness, usually minimizing some quantity.
One measure that has been recently been used is number of cut edges. In this formulation of redistricting, one is given atomic regions out of which each district must be built (e.g., in the U.S., census blocks). The populations of the atomic regions are given. Consider the graph with one vertex per atomic region and an edge between atomic regions with a shared boundary of positive length. Define the weight of a vertex to be the population of the corresponding region. A districting plan is a partition of vertices into k pieces so that the parts have nearly equal weights and each part is connected. The districts are considered compact to the extent that the plan minimizes the number of edges crossing between different parts.
There are two natural computational problems: find the most compact districting plan, and sample districting plans (possibly under a compactness constraint) uniformly at random.
Both problems are NP-hard so we consider restricting the input graph to have branchwidth at most w. (A planar graph’s branchwidth is bounded, for example, by its diameter.) If both k and w are bounded by constants, the problems are solvable in polynomial time. In this paper, we give lower and upper bounds that characterize the complexity of these problems in terms of parameters k and w. For simplicity of notation, assume that each vertex has unit weight. We would ideally like algorithms whose running times are of the form O(f(k,w) n^c) for some constant c independent of k and w (in which case the problems are said to be fixed-parameter tractable with respect to those parameters). We show that, under standard complexity-theoretic assumptions, no such algorithms exist. However, the problems are fixed-parameter tractable with respect to each of these parameters individually: there exist algorithms with running times of the form O(f(k) n^{O(w)}) and O(f(w) n^{k+1}). The first result was previously known. The new one, however, is more relevant to the application to redistricting, at least for coarse instances. Indeed, we have implemented a version of the algorithm and have used to successfully find optimally compact solutions to all redistricting instances for France (except Paris, which operates under different rules) under various population-balance constraints. For these instances, the values for w are modest and the values for k are very small.

Vincent Cohen-Addad, Philip N. Klein, Dániel Marx, Archer Wheeler, and Christopher Wolfram. On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting. In 2nd Symposium on Foundations of Responsible Computing (FORC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 192, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohenaddad_et_al:LIPIcs.FORC.2021.3, author = {Cohen-Addad, Vincent and Klein, Philip N. and Marx, D\'{a}niel and Wheeler, Archer and Wolfram, Christopher}, title = {{On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting}}, booktitle = {2nd Symposium on Foundations of Responsible Computing (FORC 2021)}, pages = {3:1--3:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-187-0}, ISSN = {1868-8969}, year = {2021}, volume = {192}, editor = {Ligett, Katrina and Gupta, Swati}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2021.3}, URN = {urn:nbn:de:0030-drops-138718}, doi = {10.4230/LIPIcs.FORC.2021.3}, annote = {Keywords: redistricting, algorithms, planar graphs, lower bounds} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We consider the k-Median problem on planar graphs: given an edge-weighted planar graph G, a set of clients C subseteq V(G), a set of facilities F subseteq V(G), and an integer parameter k, the task is to find a set of at most k facilities whose opening minimizes the total connection cost of clients, where each client contributes to the cost with the distance to the closest open facility. We give two new approximation schemes for this problem:
- FPT Approximation Scheme: for any epsilon>0, in time 2^{O(k epsilon^{-3} log (k epsilon^{-1}))}* n^O(1) we can compute a solution that has connection cost at most (1+epsilon) times the optimum, with high probability.
- Efficient Bicriteria Approximation Scheme: for any epsilon>0, in time 2^{O(epsilon^{-5} log (epsilon^{-1}))}* n^O(1) we can compute a set of at most (1+epsilon)k facilities whose opening yields connection cost at most (1+epsilon) times the optimum connection cost for opening at most k facilities, with high probability.
As a direct corollary of the second result we obtain an EPTAS for Uniform Facility Location on planar graphs, with same running time.
Our main technical tool is a new construction of a "coreset for facilities" for k-Median in planar graphs: we show that in polynomial time one can compute a subset of facilities F_0 subseteq F of size k * (log n/epsilon)^O(epsilon^{-3}) with a guarantee that there is a (1+epsilon)-approximate solution contained in F_0.

Vincent Cohen-Addad, Marcin Pilipczuk, and Michał Pilipczuk. Efficient Approximation Schemes for Uniform-Cost Clustering Problems in Planar Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohenaddad_et_al:LIPIcs.ESA.2019.33, author = {Cohen-Addad, Vincent and Pilipczuk, Marcin and Pilipczuk, Micha{\l}}, title = {{Efficient Approximation Schemes for Uniform-Cost Clustering Problems in Planar Graphs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.33}, URN = {urn:nbn:de:0030-drops-111543}, doi = {10.4230/LIPIcs.ESA.2019.33}, annote = {Keywords: k-Median, Facility Location, Planar Graphs, Approximation Scheme} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We study the complexity of the classic capacitated k-median and k-means problems parameterized by the number of centers, k. These problems are notoriously difficult since the best known approximation bound for high dimensional Euclidean space and general metric space is Theta(log k) and it remains a major open problem whether a constant factor exists.
We show that there exists a (3+epsilon)-approximation algorithm for the capacitated k-median and a (9+epsilon)-approximation algorithm for the capacitated k-means problem in general metric spaces whose running times are f(epsilon,k) n^{O(1)}. For Euclidean inputs of arbitrary dimension, we give a (1+epsilon)-approximation algorithm for both problems with a similar running time. This is a significant improvement over the (7+epsilon)-approximation of Adamczyk et al. for k-median in general metric spaces and the (69+epsilon)-approximation of Xu et al. for Euclidean k-means.

Vincent Cohen-Addad and Jason Li. On the Fixed-Parameter Tractability of Capacitated Clustering. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohenaddad_et_al:LIPIcs.ICALP.2019.41, author = {Cohen-Addad, Vincent and Li, Jason}, title = {{On the Fixed-Parameter Tractability of Capacitated Clustering}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.41}, URN = {urn:nbn:de:0030-drops-106171}, doi = {10.4230/LIPIcs.ICALP.2019.41}, annote = {Keywords: approximation algorithms, fixed-parameter tractability, capacitated, k-median, k-means, clustering, core-sets, Euclidean} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We investigate the fine-grained complexity of approximating the classical k-Median/k-Means clustering problems in general metric spaces. We show how to improve the approximation factors to (1+2/e+epsilon) and (1+8/e+epsilon) respectively, using algorithms that run in fixed-parameter time. Moreover, we show that we cannot do better in FPT time, modulo recent complexity-theoretic conjectures.

Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight FPT Approximations for k-Median and k-Means. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohenaddad_et_al:LIPIcs.ICALP.2019.42, author = {Cohen-Addad, Vincent and Gupta, Anupam and Kumar, Amit and Lee, Euiwoong and Li, Jason}, title = {{Tight FPT Approximations for k-Median and k-Means}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {42:1--42:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.42}, URN = {urn:nbn:de:0030-drops-106182}, doi = {10.4230/LIPIcs.ICALP.2019.42}, annote = {Keywords: approximation algorithms, fixed-parameter tractability, k-median, k-means, clustering, core-sets} }

Document

**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem.
A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We prove a time lower bound for this problem of n^{Omega(g/log g)} conditionally to ETH. In other words, the first n^{O(g)}-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors.
A multiway cut of an undirected graph G with t distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n^{Omega(sqrt{gt + g^2}/log(gt))}, conditionally to ETH, for any choice of the genus g >=0 of the graph and the number of terminals t >=4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case).
Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.

Vincent Cohen-Addad, Éric Colin de Verdière, Dániel Marx, and Arnaud de Mesmay. Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2019.27, author = {Cohen-Addad, Vincent and Colin de Verdi\`{e}re, \'{E}ric and Marx, D\'{a}niel and de Mesmay, Arnaud}, title = {{Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.27}, URN = {urn:nbn:de:0030-drops-104311}, doi = {10.4230/LIPIcs.SoCG.2019.27}, annote = {Keywords: Cut graph, Multiway cut, Surface, Lower bound, Parameterized Complexity, Exponential Time Hypothesis} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

In this paper we develop streaming algorithms for the diameter problem and the k-center clustering problem in the sliding window model. In this model we are interested in maintaining a solution for the N most recent points of the stream. In the diameter problem we would like to maintain two points whose distance approximates the diameter of the point set in the window. Our algorithm computes a (3 + epsilon)-approximation and uses O(1/epsilon*ln(alpha)) memory cells, where alpha is the ratio of the largest and smallest distance and is assumed to be known in advance. We also prove that under reasonable assumptions obtaining a (3 - epsilon)-approximation requires Omega(N1/3) space.
For the k-center problem, where the goal is to find k centers that minimize the maximum distance of a point to its nearest center, we obtain a (6 + epsilon)-approximation using O(k/epsilon*ln(alpha)) memory cells and a (4 + epsilon)-approximation for the special case k = 2. We also prove that any algorithm for the 2-center problem that achieves an approximation ratio of less than 4 requires Omega(N^{1/3}) space.

Vincent Cohen-Addad, Chris Schwiegelshohn, and Christian Sohler. Diameter and k-Center in Sliding Windows. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 19:1-19:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cohenaddad_et_al:LIPIcs.ICALP.2016.19, author = {Cohen-Addad, Vincent and Schwiegelshohn, Chris and Sohler, Christian}, title = {{Diameter and k-Center in Sliding Windows}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {19:1--19:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.19}, URN = {urn:nbn:de:0030-drops-63401}, doi = {10.4230/LIPIcs.ICALP.2016.19}, annote = {Keywords: Streaming, k-Center, Diameter, Sliding Windows} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude 1/epsilon^c is an approximation scheme for the following problems in the Euclidean plane: TSP with random inputs, Steiner tree with random inputs, uniform facility location (with worst case inputs), and bicriteria k-median (also with worst case inputs). The randomness assumption is necessary for TSP.

Vincent Cohen-Addad and Claire Mathieu. Effectiveness of Local Search for Geometric Optimization. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 329-344, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cohenaddad_et_al:LIPIcs.SOCG.2015.329, author = {Cohen-Addad, Vincent and Mathieu, Claire}, title = {{Effectiveness of Local Search for Geometric Optimization}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {329--344}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.329}, URN = {urn:nbn:de:0030-drops-51241}, doi = {10.4230/LIPIcs.SOCG.2015.329}, annote = {Keywords: Local Search, PTAS, Facility Location, k-Median, TSP, Steiner Tree} }

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