Ames, Brendan P. W. and Vavasis, Stephen A. (2014) Convex optimization for the planted kdisjointclique problem. Mathematical Programming, 143 (12). pp. 299337. ISSN 00255610. doi:10.1007/s1010701307331. https://resolver.caltech.edu/CaltechAUTHORS:20140213094722877

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Abstract
We consider the k disjointclique problem. The input is an undirected graph G in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph k disjoint cliques that cover the maximum number of nodes of G. This problem may be understood as a general way to pose the classical ‘clustering’ problem. In clustering, one is given data items and a distance function, and one wishes to partition the data into disjoint clusters of data items, such that the items in each cluster are close to each other. Our formulation additionally allows ‘noise’ nodes to be present in the input data that are not part of any of the cliques. The k disjointclique problem is NPhard, but we show that a convex relaxation can solve it in polynomial time for input instances constructed in a certain way. The input instances for which our algorithm finds the optimal solution consist of k disjoint large cliques (called ‘planted cliques’) that are then obscured by noise edges inserted either at random or by an adversary, as well as additional nodes not belonging to any of the k planted cliques.
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Additional Information:  © 2013 SpringerVerlag Berlin Heidelberg and Mathematical Optimization Society. Received: 20 August 2010; Accepted: 9 October 2012; Published online: 7 December 2013. Supported in part by a Discovery Grant and Postgraduate Scholarship (Doctoral) from NSERC (Natural Science and Engineering Research Council of Canada), MITACS (Mathematics of Information Technology and Complex Systems), and the US Air Force Office of Scientific Research.  
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Issue or Number:  12  
Classification Code:  Mathematics Subject Classification: 90C22; 90C25; 68Q25; 62H302  
DOI:  10.1007/s1010701307331  
Record Number:  CaltechAUTHORS:20140213094722877  
Persistent URL:  https://resolver.caltech.edu/CaltechAUTHORS:20140213094722877  
Official Citation:  Ames, B. W., & Vavasis, S. (2014). Convex optimization for the planted kdisjointclique problem. Mathematical Programming, 143(12), 299337. doi: 10.1007/s1010701307331  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  43808  
Collection:  CaltechAUTHORS  
Deposited By:  Jason Perez  
Deposited On:  13 Feb 2014 18:12  
Last Modified:  10 Nov 2021 16:43 
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