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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.58
Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance

Authors: Vikrant Ashvinkumar, Sepehr Assadi, Chengyuan Deng, Jie Gao, and Chen Wang

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

Abstract
Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced. We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation. To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest.
Cite as

Vikrant Ashvinkumar, Sepehr Assadi, Chengyuan Deng, Jie Gao, and Chen Wang. Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ashvinkumar_et_al:LIPIcs.APPROX/RANDOM.2023.58,
  author =	{Ashvinkumar, Vikrant and Assadi, Sepehr and Deng, Chengyuan and Gao, Jie and Wang, Chen},
  title =	{{Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{58:1--58:23},
  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.2023.58},
  URN =		{urn:nbn:de:0030-drops-188830},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.58},
  annote =	{Keywords: Streaming algorithms, structural balance, pseudo-randomness generator}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.2
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

Authors: Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.
Cite as

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2,
  author =	{Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung},
  title =	{{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2},
  URN =		{urn:nbn:de:0030-drops-160109},
  doi =		{10.4230/LIPIcs.SoCG.2022.2},
  annote =	{Keywords: Approximation, Motion Planning, Scheduling}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.7
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Authors: Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/DagSemProc.07151.2
Discovery of Sensor Network Layout using Connectivity Information

Authors: Jie Gao, Sol Lederer, and Yue Wang

Published in: Dagstuhl Seminar Proceedings, Volume 7151, Geometry in Sensor Networks (2007)

Abstract
We propose a distributed algorithm to discover and recover the layout of a large sensor network having a complex shape. As sensor network deployments grow large in size and become non-uniform, localization algorithms suffer from ``flip'' ambiguities---where a part of the network folds on top of another while keeping all edge length measurements preserved. We explore the high-order topological information in a sensor field to prevent incorrect flips and accurately recover the shape of the sensor network. We select landmarks on network boundaries with sufficient density, construct the landmark Voronoi diagram and its dual combinatorial Delaunay complex on these landmarks. The key insight is that when the landmarks are dense enough to capture the local geometric complexity, the combinatorial Delaunay complex is globally rigid and has a unique realization in the plane. An embedding by simply gluing the Delaunay triangles properly derives a faithful network layout, which consequently leads to a practical and sufficiently accurate localization algorithm. We prove the global rigidity of the combinatorial Delaunay complex in the case of a continuous geometric region. Simulation results on discrete networks show surprisingly good results, while multi-dimensional scaling and rubberband representation perform poorly or not at all in recovering the network layout. This is joint work with Sol Lederer and Yue Wang.
Cite as

Jie Gao, Sol Lederer, and Yue Wang. Discovery of Sensor Network Layout using Connectivity Information. In Geometry in Sensor Networks. Dagstuhl Seminar Proceedings, Volume 7151, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{gao_et_al:DagSemProc.07151.2,
  author =	{Gao, Jie and Lederer, Sol and Wang, Yue},
  title =	{{Discovery of Sensor Network Layout using Connectivity Information}},
  booktitle =	{Geometry in Sensor Networks},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7151},
  editor =	{Subhash Suri and Roger Wattenhofer and Peter Widmayer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.07151.2},
  URN =		{urn:nbn:de:0030-drops-11149},
  doi =		{10.4230/DagSemProc.07151.2},
  annote =	{Keywords: Sensor Networks, Localization, Delaunay complex, Rigidity}
}
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