Manghiuc, B.-A., Peng, P. orcid.org/0000-0003-2700-5699 and Sun, H. (2020) Augmenting the algebraic connectivity of graphs. In: Grandoni, F., Herman, G. and Sanders, P., (eds.) 28th Annual European Symposium on Algorithms (ESA 2020). 28th European Symposium on Algorithms (ESA 2020), 07-10 Sep 2020, Online conference. Leibniz International Proceedings in Informatics (LIPIcs) . Schloss Dagstuhl--Leibniz-Zentrum fur Informatik , 70:1-70:22. ISBN 9783959771627
Abstract
For any undirected graph G=(V,E) and a set EW of candidate edges with E∩EW=∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from EW with appropriate weighting, such that the algebraic connectivity of the resulting graph H=(V,E∪F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph's conductance and the mixing time of random walks in a graph, maximising the resulting graph's algebraic connectivity by adding a small number of edges has been studied over the past 15 years. In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem. (1) We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [GB06]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly. (2) We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n2mk) time [KMST10]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2020 The Authors. Open access. Licensed under Creative Commons License CC-BY (https://creativecommons.org/licenses/by/3.0/). |
Keywords: | Graph sparsification; Algebraic connectivity; Semidefinite programming |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 01 Jul 2020 07:50 |
Last Modified: | 21 Oct 2020 13:11 |
Status: | Published |
Publisher: | Schloss Dagstuhl--Leibniz-Zentrum fur Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.ESA.2020.70 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:162667 |