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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

The problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion O(c²), where c denotes the optimal distortion [Bădoiu et al. 2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of outliers.
Specifically, we say that a metric space (X,ρ) admits a (k,c)-embedding if there exists K ⊂ X, with |K| = k, such that (X⧵ K, ρ) admits an embedding into the line with distortion at most c. Given k ≥ 0, and a metric space that admits a (k,c)-embedding, for some c ≥ 1, our algorithm computes a (poly(k, c, log n), poly(c))-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion.

Karine Chubarian and Anastasios Sidiropoulos. Computing Bi-Lipschitz Outlier Embeddings into the Line. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chubarian_et_al:LIPIcs.APPROX/RANDOM.2020.36, author = {Chubarian, Karine and Sidiropoulos, Anastasios}, title = {{Computing Bi-Lipschitz Outlier Embeddings into the Line}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {36:1--36:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, 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.2020.36}, URN = {urn:nbn:de:0030-drops-126398}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.36}, annote = {Keywords: metric embeddings, outliers, distortion, approximation algorithms} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We study the problem of finding a mapping f from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points (u,v,w) asserts that |f(u)-f(v)| < |f(u)-f(w)|. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies (1-ε)-fraction of all constraints, our algorithm computes a solution that satisfies (1-O(ε^{1/8}))-fraction of all constraints, in time O(n⁷) + (1/ε)^{O(1/ε^{1/8})} n.

Bohan Fan, Diego Ihara, Neshat Mohammadi, Francesco Sgherzi, Anastasios Sidiropoulos, and Mina Valizadeh. Learning Lines with Ordinal Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.APPROX/RANDOM.2020.45, author = {Fan, Bohan and Ihara, Diego and Mohammadi, Neshat and Sgherzi, Francesco and Sidiropoulos, Anastasios and Valizadeh, Mina}, title = {{Learning Lines with Ordinal Constraints}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, 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.2020.45}, URN = {urn:nbn:de:0030-drops-126486}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.45}, annote = {Keywords: metric learning, embedding into the line, ordinal constraints, approximation algorithms} }

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APPROX

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

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016].
Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands.

Timothy Carpenter, Ario Salmasi, and Anastasios Sidiropoulos. Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carpenter_et_al:LIPIcs.APPROX-RANDOM.2019.14, author = {Carpenter, Timothy and Salmasi, Ario and Sidiropoulos, Anastasios}, title = {{Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {14:1--14:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.14}, URN = {urn:nbn:de:0030-drops-112290}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.14}, annote = {Keywords: Routing, Node-disjoint, Symmetric demands, Minor-free graphs} }

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

We study the problem of supervised learning a metric space under discriminative constraints. Given a universe X and sets S, D subset binom{X}{2} of similar and dissimilar pairs, we seek to find a mapping f:X -> Y, into some target metric space M=(Y,rho), such that similar objects are mapped to points at distance at most u, and dissimilar objects are mapped to points at distance at least l. More generally, the goal is to find a mapping of maximum accuracy (that is, fraction of correctly classified pairs). We propose approximation algorithms for various versions of this problem, for the cases of Euclidean and tree metric spaces. For both of these target spaces, we obtain fully polynomial-time approximation schemes (FPTAS) for the case of perfect information. In the presence of imperfect information we present approximation algorithms that run in quasi-polynomial time (QPTAS). We also present an exact algorithm for learning line metric spaces with perfect information in polynomial time. Our algorithms use a combination of tools from metric embeddings and graph partitioning, that could be of independent interest.

Diego Ihara, Neshat Mohammadi, and Anastasios Sidiropoulos. Algorithms for Metric Learning via Contrastive Embeddings. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ihara_et_al:LIPIcs.SoCG.2019.45, author = {Ihara, Diego and Mohammadi, Neshat and Sidiropoulos, Anastasios}, title = {{Algorithms for Metric Learning via Contrastive Embeddings}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {45:1--45:14}, 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.2019.45}, URN = {urn:nbn:de:0030-drops-104492}, doi = {10.4230/LIPIcs.SoCG.2019.45}, annote = {Keywords: metric learning, contrastive distortion, embeddings, algorithms} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give
- an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c);
- an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

Timothy Carpenter, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Anastasios Sidiropoulos. Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carpenter_et_al:LIPIcs.SoCG.2018.21, author = {Carpenter, Timothy and Fomin, Fedor V. and Lokshtanov, Daniel and Saurabh, Saket and Sidiropoulos, Anastasios}, title = {{Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.21}, URN = {urn:nbn:de:0030-drops-87344}, doi = {10.4230/LIPIcs.SoCG.2018.21}, annote = {Keywords: Metric embeddings, minimum-distortion embeddings, 1-dimensional simplicial complex, Fixed-parameter tractable algorithms, Approximation algorithms} }

Document

**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.
More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:
- For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}.
- For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014].

Anastasios Sidiropoulos, Kritika Singhal, and Vijay Sridhar. Fractal Dimension and Lower Bounds for Geometric Problems. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{sidiropoulos_et_al:LIPIcs.SoCG.2018.70, author = {Sidiropoulos, Anastasios and Singhal, Kritika and Sridhar, Vijay}, title = {{Fractal Dimension and Lower Bounds for Geometric Problems}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.70}, URN = {urn:nbn:de:0030-drops-87831}, doi = {10.4230/LIPIcs.SoCG.2018.70}, annote = {Keywords: fractal dimension, treewidth, spanners, lower bounds, exponential time hypothesis} }

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

We study hierarchical clusterings of metric spaces that change over time. This is a natural geo- metric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric space, which naturally induces a hierarchical clustering of the set of points. We enforce temporal coherence among the embeddings by finding correspondences between successive pairs of ultrametric spaces which exhibit small distortion in the Gromov-Hausdorff sense. We present both upper and lower bounds on the approximability of the resulting optimization problems.

Tamal K. Dey, Alfred Rossi, and Anastasios Sidiropoulos. Temporal Hierarchical Clustering. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dey_et_al:LIPIcs.ISAAC.2017.28, author = {Dey, Tamal K. and Rossi, Alfred and Sidiropoulos, Anastasios}, title = {{Temporal Hierarchical Clustering}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {28:1--28:12}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.28}, URN = {urn:nbn:de:0030-drops-82519}, doi = {10.4230/LIPIcs.ISAAC.2017.28}, annote = {Keywords: clustering, hierarchical clustering, multi-objective optimization, dynamic metric spaces, moving point sets, approximation algorithms} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i.e. static) clustering problems are not applicable anymore.
We propose a set of optimization problems which we collectively refer to as temporal clustering. The quality of a solution to a temporal clustering instance can be quantified using three parameters: the number of clusters k, the spatial clustering cost r, and the maximum cluster displacement delta between consecutive time steps. We consider spatial clustering costs which generalize the well-studied k-center, discrete k-median, and discrete k-means objectives of classical clustering problems. We develop new algorithms that achieve trade-offs between the three objectives k, r, and delta. Our upper bounds are complemented by inapproximability results.

Tamal K. Dey, Alfred Rossi, and Anastasios Sidiropoulos. Temporal Clustering. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dey_et_al:LIPIcs.ESA.2017.34, author = {Dey, Tamal K. and Rossi, Alfred and Sidiropoulos, Anastasios}, title = {{Temporal Clustering}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.34}, URN = {urn:nbn:de:0030-drops-78567}, doi = {10.4230/LIPIcs.ESA.2017.34}, annote = {Keywords: clustering, multi-objective optimization, dynamic metric spaces, moving point sets, approximation algorithms, hardness of approximation} }

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

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta).
Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.
We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension.
However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space.

Anastasios Sidiropoulos and Vijay Sridhar. Algorithmic Interpretations of Fractal Dimension. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{sidiropoulos_et_al:LIPIcs.SoCG.2017.58, author = {Sidiropoulos, Anastasios and Sridhar, Vijay}, title = {{Algorithmic Interpretations of Fractal Dimension}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {58:1--58:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.58}, URN = {urn:nbn:de:0030-drops-72126}, doi = {10.4230/LIPIcs.SoCG.2017.58}, annote = {Keywords: fractal dimension, exact algorithms, fixed parameter tractability, approximation schemes, spanners} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.
We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).

Arturs Backurs and Anastasios Sidiropoulos. Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{backurs_et_al:LIPIcs.APPROX-RANDOM.2016.1, author = {Backurs, Arturs and Sidiropoulos, Anastasios}, title = {{Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l\underlinep Spaces}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, 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.2016.1}, URN = {urn:nbn:de:0030-drops-66241}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.1}, annote = {Keywords: metric embeddings, Hausdorff metric, distortion, dimension} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].
We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.

Dániel Marx, Ario Salmasi, and Anastasios Sidiropoulos. Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 16:1-16:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{marx_et_al:LIPIcs.APPROX-RANDOM.2016.16, author = {Marx, D\'{a}niel and Salmasi, Ario and Sidiropoulos, Anastasios}, title = {{Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {16:1--16:54}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, 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.2016.16}, URN = {urn:nbn:de:0030-drops-66391}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.16}, annote = {Keywords: asymmetric TSP, approximation algorithms, nearly-embeddable graphs, Held-Karp LP, exponential time hypothesis} }

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

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces.
Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t.
The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

Facundo Mémoli, Anastasios Sidiropoulos, and Vijay Sridhar. Quasimetric Embeddings and Their Applications. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{memoli_et_al:LIPIcs.ICALP.2016.85, author = {M\'{e}moli, Facundo and Sidiropoulos, Anastasios and Sridhar, Vijay}, title = {{Quasimetric Embeddings and Their Applications}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {85:1--85:14}, 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.2016.85}, URN = {urn:nbn:de:0030-drops-62007}, doi = {10.4230/LIPIcs.ICALP.2016.85}, annote = {Keywords: metric embeddings, quasimetrics, outliers, random embeddings, treewidth, Directed Sparsest-Cut, Directed Multicut} }

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