Smoothed analysis on connected graphs

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edges of G at random (usually adding few random edges to G) typically results in a graph having much "nicer" properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an n-vertex connected graph G, form a random supergraph G∗of G by turning every pair of vertices of G into an edge with probability ∈/n , where ∈ is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have very large diameter, and possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph then typically G∗ has edge expansion Ω(1/log n ), diameter O(1/log n), vertex expansion ( 1/log n ), and contains a path of length (n), where for the last two properties we additionally assume that G has bounded maximum degree. Moreover, we show that if G has bounded degeneracy, then typically the mixing time of the lazy random walk on G∗ is O(log2 n). All these results are asymptotically tight.

Original languageEnglish
Title of host publicationLeibniz International Proceedings in Informatics, LIPIcs
EditorsKlaus Jansen, Jose D. P. Rolim, Nikhil R. Devanur, Cristopher Moore
Pages810-825
Number of pages16
ISBN (Electronic)9783939897743
DOIs
StatePublished - 1 Sep 2014
Event17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014 - Barcelona, Spain
Duration: 4 Sep 20146 Sep 2014

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume28

Conference

Conference17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014
Country/TerritorySpain
CityBarcelona
Period4/09/146/09/14

Keywords

  • Random network models
  • Random walks and Markov chains

All Science Journal Classification (ASJC) codes

  • Software

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