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 -