2 Search Results for "Galhotra, Sainyam"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.53
Connectivity of Random Annulus Graphs and the Geometric Block Model

Authors: Sainyam Galhotra, Arya Mazumdar, Soumyabrata Pal, and Barna Saha

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

Abstract
Random geometric graph (Gilbert, 1961) is a basic model of random graphs for spatial networks proposed shortly after the introduction of the Erdős-Rényi random graphs. The geometric block model (GBM) is a probabilistic model for community detection defined over random geometric graphs (RGG) similar in spirit to the popular stochastic block model which is defined over Erdős-Rényi random graphs. The GBM naturally inherits many desirable properties of RGGs such as transitivity ("friends having common friends') and has been shown to model many real-world networks better than the stochastic block model. Analyzing the properties of a GBM requires new tools and perspectives to handle correlation in edge formation. In this paper, we study the necessary and sufficient conditions for community recovery over GBM in the connectivity regime. We provide efficient algorithms that recover the communities exactly with high probability and match the lower bound within a small constant factor. This requires us to prove new connectivity results for vertex-random graphs or random annulus graphs which are natural generalizations of random geometric graphs. A vertex-random graph is a model of random graphs where the randomness lies in the vertices as opposed to an Erdős-Rényi random graph where the randomness lies in the edges. A vertex-random graph G(n, [r_1, r_2]), 0 <=r_1 <r_2 <=1 with n vertices is defined by assigning a real number in [0,1] randomly and uniformly to each vertices and adding an edge between two vertices if the "distance" between the corresponding two random numbers is between r_1 and r_2. For the special case of r_1=0, this corresponds to random geometric graph in one dimension. We can extend this model naturally to higher dimensions; these higher dimensional counterparts are referred to as random annulus graphs. Random annulus graphs appear naturally whenever the well-known Goldilocks principle ("not too close, not too far') holds in a network. In this paper, we study the connectivity properties of such graphs, providing both necessary and sufficient conditions. We show a surprising long edge phenomena for vertex-random graphs: the minimum gap for connectivity between r_1 and r_2 is significantly less when r_1 >0 vs when r_1=0 (RGG). We then extend the connectivity results to high dimensions. These results play a crucial role in analyzing the GBM.
Cite as

Sainyam Galhotra, Arya Mazumdar, Soumyabrata Pal, and Barna Saha. Connectivity of Random Annulus Graphs and the Geometric Block Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 53:1-53:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galhotra_et_al:LIPIcs.APPROX-RANDOM.2019.53,
  author =	{Galhotra, Sainyam and Mazumdar, Arya and Pal, Soumyabrata and Saha, Barna},
  title =	{{Connectivity of Random Annulus Graphs and the Geometric Block Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{53:1--53:23},
  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.2019.53},
  URN =		{urn:nbn:de:0030-drops-112682},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.53},
  annote =	{Keywords: random graphs, geometric graphs, community detection, block model}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.81
Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost

Authors: Barna Saha and Sanjay Subramanian

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

Abstract
Several clustering frameworks with interactive (semi-supervised) queries have been studied in the past. Recently, clustering with same-cluster queries has become popular. An algorithm in this setting has access to an oracle with full knowledge of an optimal clustering, and the algorithm can ask the oracle queries of the form, "Does the optimal clustering put vertices u and v in the same cluster?" Due to its simplicity, this querying model can easily be implemented in real crowd-sourcing platforms and has attracted a lot of recent work. In this paper, we study the popular correlation clustering problem (Bansal et al., 2002) under the same-cluster querying framework. Given a complete graph G=(V,E) with positive and negative edge labels, correlation clustering objective aims to compute a graph clustering that minimizes the total number of disagreements, that is the negative intra-cluster edges and positive inter-cluster edges. In a recent work, Ailon et al. (2018b) provided an approximation algorithm for correlation clustering that approximates the correlation clustering objective within (1+epsilon) with O((k^{14} log{n} log{k})/epsilon^6) queries when the number of clusters, k, is fixed. For many applications, k is not fixed and can grow with |V|. Moreover, the dependency of k^14 on query complexity renders the algorithm impractical even for datasets with small values of k. In this paper, we take a different approach. Let C_{OPT} be the number of disagreements made by the optimal clustering. We present algorithms for correlation clustering whose error and query bounds are parameterized by C_{OPT} rather than by the number of clusters. Indeed, a good clustering must have small C_{OPT}. Specifically, we present an efficient algorithm that recovers an exact optimal clustering using at most 2C_{OPT} queries and an efficient algorithm that outputs a 2-approximation using at most C_{OPT} queries. In addition, we show under a plausible complexity assumption, there does not exist any polynomial time algorithm that has an approximation ratio better than 1+alpha for an absolute constant alpha > 0 with o(C_{OPT}) queries. Therefore, our first algorithm achieves the optimal query bound within a factor of 2. We extensively evaluate our methods on several synthetic and real-world datasets using real crowd-sourced oracles. Moreover, we compare our approach against known correlation clustering algorithms that do not perform querying. In all cases, our algorithms exhibit superior performance.
Cite as

Barna Saha and Sanjay Subramanian. Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 81:1-81:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{saha_et_al:LIPIcs.ESA.2019.81,
  author =	{Saha, Barna and Subramanian, Sanjay},
  title =	{{Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{81:1--81:17},
  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.2019.81},
  URN =		{urn:nbn:de:0030-drops-112020},
  doi =		{10.4230/LIPIcs.ESA.2019.81},
  annote =	{Keywords: Clustering, Approximation Algorithm, Crowdsourcing, Randomized Algorithm}
}
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