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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Given n pairs of points, S = {{p_1, q_1}, {p_2, q_2}, ..., {p_n, q_n}}, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Su Jia, Matthew J. Katz, Tyler Mayer, and Joseph S. B. Mitchell. Network Optimization on Partitioned Pairs of Points. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arkin_et_al:LIPIcs.ISAAC.2017.6, author = {Arkin, Esther M. and Banik, Aritra and Carmi, Paz and Citovsky, Gui and Jia, Su and Katz, Matthew J. and Mayer, Tyler and Mitchell, Joseph S. B.}, title = {{Network Optimization on Partitioned Pairs of Points}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {6:1--6:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.6}, URN = {urn:nbn:de:0030-drops-82700}, doi = {10.4230/LIPIcs.ISAAC.2017.6}, annote = {Keywords: Network Optimization, TSP tour, Matching, Spanning Tree, Pairs, Partition, Algorithms, Complexity} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space (X, d) and a set of subsets R = {R_1, R_2, ... , R_n} : R_i subseteq X, the goal is to devise an ordering of the regions, sigma_R, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by sigma_R is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the adversarial model in which once the choice of sigma_R is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions R that is optimal with respect to the ``worst'' point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a 3-approximation when R is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which R is a set of disjoint unit disks in the plane.

Gui Citovsky, Tyler Mayer, and Joseph S. B. Mitchell. TSP With Locational Uncertainty: The Adversarial Model. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{citovsky_et_al:LIPIcs.SoCG.2017.32, author = {Citovsky, Gui and Mayer, Tyler and Mitchell, Joseph S. B.}, title = {{TSP With Locational Uncertainty: The Adversarial Model}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.32}, URN = {urn:nbn:de:0030-drops-72334}, doi = {10.4230/LIPIcs.SoCG.2017.32}, annote = {Keywords: traveling salesperson problem, TSP with neighborhoods, approximation algorithms, uncertainty} }

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