Found 3 Possible Name Variants:

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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Given a geometric domain P, visibility-based search problems seek routes for one or more mobile agents ("watchmen") to move within P in order to be able to see a portion (or all) of P, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within P according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of P. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain P in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of P (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon P we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains P (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in P to guarantee some specified probability of detection of a static target within P, randomly distributed in P according to a given prior distribution.

Kien C. Huynh, Joseph S. B. Mitchell, Linh Nguyen, and Valentin Polishchuk. Optimizing Visibility-Based Search in Polygonal Domains. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{huynh_et_al:LIPIcs.SWAT.2024.27, author = {Huynh, Kien C. and Mitchell, Joseph S. B. and Nguyen, Linh and Polishchuk, Valentin}, title = {{Optimizing Visibility-Based Search in Polygonal Domains}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.27}, URN = {urn:nbn:de:0030-drops-200671}, doi = {10.4230/LIPIcs.SWAT.2024.27}, annote = {Keywords: Quota watchman route problem, budgeted watchman route problem, visibility-based search, approximation} }

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

The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min - the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible.
We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs.
We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk. On Flipping the Fréchet Distance. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.51, author = {Filtser, Omrit and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin}, title = {{On Flipping the Fr\'{e}chet Distance}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {51:1--51:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.51}, URN = {urn:nbn:de:0030-drops-175548}, doi = {10.4230/LIPIcs.ITCS.2023.51}, annote = {Keywords: curves, polygons, distancing measure} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2 λ.

A. Karim Abu-Affash, Sujoy Bhore, Paz Carmi, and Joseph S. B. Mitchell. Planar Bichromatic Bottleneck Spanning Trees. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{abuaffash_et_al:LIPIcs.ESA.2020.1, author = {Abu-Affash, A. Karim and Bhore, Sujoy and Carmi, Paz and Mitchell, Joseph S. B.}, title = {{Planar Bichromatic Bottleneck Spanning Trees}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.1}, URN = {urn:nbn:de:0030-drops-128670}, doi = {10.4230/LIPIcs.ESA.2020.1}, annote = {Keywords: Approximation Algorithms, Bottleneck Spanning Tree, NP-Hardness} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.

Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arkin_et_al:LIPIcs.ESA.2020.7, author = {Arkin, Esther M. and Das, Rathish and Gao, Jie and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin and T\'{o}th, Csaba D.}, title = {{Cutting Polygons into Small Pieces with Chords: Laser-Based Localization}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {7:1--7:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.7}, URN = {urn:nbn:de:0030-drops-128736}, doi = {10.4230/LIPIcs.ESA.2020.7}, annote = {Keywords: Polygon partition, Arrangements, Visibility, Localization} }

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**Published in:** LIPIcs, Volume 160, 18th International Symposium on Experimental Algorithms (SEA 2020)

We study a trajectory analysis problem we call the Trajectory Capture Problem (TCP), in which, for a given input set T of trajectories in the plane, and an integer k≥ 2, we seek to compute a set of k points ("portals") to maximize the total weight of all subtrajectories of T between pairs of portals. This problem naturally arises in trajectory analysis and summarization.
We show that the TCP is NP-hard (even in very special cases) and give some first approximation results. Our main focus is on attacking the TCP with practical algorithm-engineering approaches, including integer linear programming (to solve instances to provable optimality) and local search methods. We study the integrality gap arising from such approaches. We analyze our methods on different classes of data, including benchmark instances that we generate. Our goal is to understand the best performing heuristics, based on both solution time and solution quality. We demonstrate that we are able to compute provably optimal solutions for real-world instances.

Sándor P. Fekete, Alexander Hill, Dominik Krupke, Tyler Mayer, Joseph S. B. Mitchell, Ojas Parekh, and Cynthia A. Phillips. Probing a Set of Trajectories to Maximize Captured Information. In 18th International Symposium on Experimental Algorithms (SEA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 160, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fekete_et_al:LIPIcs.SEA.2020.5, author = {Fekete, S\'{a}ndor P. and Hill, Alexander and Krupke, Dominik and Mayer, Tyler and Mitchell, Joseph S. B. and Parekh, Ojas and Phillips, Cynthia A.}, title = {{Probing a Set of Trajectories to Maximize Captured Information}}, booktitle = {18th International Symposium on Experimental Algorithms (SEA 2020)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-148-1}, ISSN = {1868-8969}, year = {2020}, volume = {160}, editor = {Faro, Simone and Cantone, Domenico}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2020.5}, URN = {urn:nbn:de:0030-drops-120796}, doi = {10.4230/LIPIcs.SEA.2020.5}, annote = {Keywords: Algorithm engineering, optimization, complexity, approximation, trajectories} }

Document

**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π=f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p∈P and q∈P, the length of the sub-curve of π connecting f(p) with f(q) is at most β||f(p)-f(q)‖, where ‖.‖ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. The βSP also asks that we output P if it exists.
The βSP quantifies a notion of "near straightness" for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes.
We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε>0, β∈O(poly(log |V(G)|)), and s,t∈V(G), outputs a β-stretch path between s and t, if a (1-ε)β-stretch path between s and t exists in the drawing.

Esther M. Arkin, Faryad Darabi Sahneh, Alon Efrat, Fabian Frank, Radoslav Fulek, Stephen Kobourov, and Joseph S. B. Mitchell. Computing β-Stretch Paths in Drawings of Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arkin_et_al:LIPIcs.SWAT.2020.7, author = {Arkin, Esther M. and Sahneh, Faryad Darabi and Efrat, Alon and Frank, Fabian and Fulek, Radoslav and Kobourov, Stephen and Mitchell, Joseph S. B.}, title = {{Computing \beta-Stretch Paths in Drawings of Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.7}, URN = {urn:nbn:de:0030-drops-122540}, doi = {10.4230/LIPIcs.SWAT.2020.7}, annote = {Keywords: stretch factor, dilation, geometric spanners} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Given a set D of n unit disks in the plane and an integer k <= n, the maximum area connected subset problem asks for a set D' subseteq D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint.
We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 1/2-approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of epsilon k disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation.

Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Joseph S. B. Mitchell, and Nabil H. Mustafa. Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{huang_et_al:LIPIcs.APPROX-RANDOM.2019.32, author = {Huang, Chien-Chung and Mari, Mathieu and Mathieu, Claire and Mitchell, Joseph S. B. and Mustafa, Nabil H.}, title = {{Maximizing Covered Area in the Euclidean Plane with Connectivity Constraint}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {32:1--32:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.32}, URN = {urn:nbn:de:0030-drops-112471}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.32}, annote = {Keywords: approximation algorithm, submodular function optimisation, unit disk graph, connectivity constraint} }

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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} }

Document

**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.

Sándor P. Fekete, Qian Li, Joseph S. B. Mitchell, and Christian Scheffer. Universal Guard Problems. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fekete_et_al:LIPIcs.ISAAC.2016.32, author = {Fekete, S\'{a}ndor P. and Li, Qian and Mitchell, Joseph S. B. and Scheffer, Christian}, title = {{Universal Guard Problems}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {32:1--32:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.32}, URN = {urn:nbn:de:0030-drops-68022}, doi = {10.4230/LIPIcs.ISAAC.2016.32}, annote = {Keywords: Art Gallery Problem, universal guarding, polygonization, worst-case bounds, robust covering} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the L_1 geodesic diameter in O(n^2+h^4) time and the L_1 geodesic center in O((n^4+n^2 h^4)*alpha(n)) time, where alpha(.) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in O(n^{7.73}) or O(n^7(h+log(n))) time, and compute the geodesic center in O(n^{12+epsilon}) time. Therefore, our algorithms are much faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on L_1 shortest paths in polygonal domains.

Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, and Haitao Wang. Computing the L1 Geodesic Diameter and Center of a Polygonal Domain. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{wonbae_et_al:LIPIcs.STACS.2016.14, author = {Won Bae, Sang and Korman, Matias and Mitchell, Joseph S. B. and Okamoto, Yoshio and Polishchuk, Valentin and Wang, Haitao}, title = {{Computing the L1 Geodesic Diameter and Center of a Polygonal Domain}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.14}, URN = {urn:nbn:de:0030-drops-57151}, doi = {10.4230/LIPIcs.STACS.2016.14}, annote = {Keywords: geodesic diameter, geodesic center, shortest paths, polygonal domains, L1 metric} }

Document

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

We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.

Esther M. Arkin, Alon Efrat, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, and Topi Talvitie. Shortest Path to a Segment and Quickest Visibility Queries. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 658-673, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{arkin_et_al:LIPIcs.SOCG.2015.658, author = {Arkin, Esther M. and Efrat, Alon and Knauer, Christian and Mitchell, Joseph S. B. and Polishchuk, Valentin and Rote, G\"{u}nter and Schlipf, Lena and Talvitie, Topi}, title = {{Shortest Path to a Segment and Quickest Visibility Queries}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {658--673}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.658}, URN = {urn:nbn:de:0030-drops-51474}, doi = {10.4230/LIPIcs.SOCG.2015.658}, annote = {Keywords: path planning, visibility, query structures and complexity, persistent data structures, continuous Dijkstra} }

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**Published in:** OASIcs, Volume 42, 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2014)

We study a facility location problem motivated by requirements pertaining to the distribution of charging stations for electric vehicles: Place a minimum number of battery charging stations at a subset of nodes of a network, so that battery-powered electric vehicles will be able to move between destinations using "t-spanning" routes, of lengths within a factor t > 1 of the length of a shortest path, while having sufficient charging stations along the way. We give constant-factor approximation algorithms for minimizing the number of charging stations, subject to the t-spanning constraint. We study two versions of the problem, one in which the stations are required to support a single ride (to a single destination), and one in which the stations are to support multiple rides through a sequence of destinations, where the destinations are revealed one at a time.

Esther M. Arkin, Paz Carmi, Matthew J. Katz, Joseph S. B. Mitchell, and Michael Segal. Locating Battery Charging Stations to Facilitate Almost Shortest Paths. In 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 42, pp. 25-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{arkin_et_al:OASIcs.ATMOS.2014.25, author = {Arkin, Esther M. and Carmi, Paz and Katz, Matthew J. and Mitchell, Joseph S. B. and Segal, Michael}, title = {{Locating Battery Charging Stations to Facilitate Almost Shortest Paths}}, booktitle = {14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems}, pages = {25--33}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-75-0}, ISSN = {2190-6807}, year = {2014}, volume = {42}, editor = {Funke, Stefan and Mihal\'{a}k, Mat\'{u}s}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2014.25}, URN = {urn:nbn:de:0030-drops-47500}, doi = {10.4230/OASIcs.ATMOS.2014.25}, annote = {Keywords: approximation algorithms; geometric spanners; transportation networks} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9111, Computational Geometry (2009)

This is a scribing of the open problems posed at the Tuesday evening open problem session. Posers of problems provided input after the session.

Joseph S. Mitchell. Open Problem Session. In Computational Geometry. Dagstuhl Seminar Proceedings, Volume 9111, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{mitchell:DagSemProc.09111.4, author = {Mitchell, Joseph S.}, title = {{Open Problem Session}}, booktitle = {Computational Geometry}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9111}, editor = {Pankaj Kumar Agarwal and Helmut Alt and Monique Teillaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09111.4}, URN = {urn:nbn:de:0030-drops-20308}, doi = {10.4230/DagSemProc.09111.4}, annote = {Keywords: Open problems, computational geometry} }

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**Published in:** OASIcs, Volume 12, 9th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'09) (2009)

We address a problem in air traffic management: scheduling flights
in order to minimize the maximum number of aircraft that
simultaneously lie within a single air traffic control sector at
any time $t$. Since the problem is a generalization of the NP-hard
no-wait job-shop scheduling, we resort to heuristics. We report
experimental results for real-world flight data.

Joondong Kim, Alexander Kroeller, and Joseph Mitchell. Scheduling Aircraft to Reduce Controller Workload. In 9th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'09). Open Access Series in Informatics (OASIcs), Volume 12, pp. -12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kim_et_al:OASIcs.ATMOS.2009.2144, author = {Kim, Joondong and Kroeller, Alexander and Mitchell, Joseph}, title = {{Scheduling Aircraft to Reduce Controller Workload}}, booktitle = {9th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'09)}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-939897-11-8}, ISSN = {2190-6807}, year = {2009}, volume = {12}, editor = {Clausen, Jens and Di Stefano, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2009.2144}, URN = {urn:nbn:de:0030-drops-21443}, doi = {10.4230/OASIcs.ATMOS.2009.2144}, annote = {Keywords: Air Traffic Management, trajectory scheduling, flight plan scheduling, no-wait job shop Air Traffic Management, trajectory scheduling, flight plan scheduling, no-wait job shop} }

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