TY - JOUR

T1 - Smoothed analysis on connected graphs

AU - Krivelevich, Michael

AU - Reichman, Daniel

AU - Samotij, Wojciech

N1 - Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.

PY - 2015

Y1 - 2015

N2 - The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edge set 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 Gâ of G by turning every pair of vertices of G into an edge with probability Îμ n, where Îμ 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 a very large diameter, and can possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph, then typically Ghas edge expansion ω( 1 log n ), diameter O(log n), and vertex expansion ω( 1 log n ) and contains a path of length ω(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 Gis O(log2 n). All these results are asymptotically tight.

AB - The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edge set 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 Gâ of G by turning every pair of vertices of G into an edge with probability Îμ n, where Îμ 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 a very large diameter, and can possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph, then typically Ghas edge expansion ω( 1 log n ), diameter O(log n), and vertex expansion ω( 1 log n ) and contains a path of length ω(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 Gis O(log2 n). All these results are asymptotically tight.

KW - Graph expansion

KW - Random walks

KW - Small worlds

UR - http://www.scopus.com/inward/record.url?scp=84943186096&partnerID=8YFLogxK

U2 - 10.1137/151002496

DO - 10.1137/151002496

M3 - מאמר

AN - SCOPUS:84943186096

VL - 29

SP - 1654

EP - 1669

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -