TY - GEN

T1 - Smoothed analysis on connected graphs

AU - Krivelevich, Michael

AU - Reichman, Daniel

AU - Samotij, Wojciech

N1 - Publisher Copyright: © Michael Krivelevich, Daniel Reichman, and Wojciech Samotij.

PY - 2014/9/1

Y1 - 2014/9/1

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 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.

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 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.

KW - Random network models

KW - Random walks and Markov chains

UR - http://www.scopus.com/inward/record.url?scp=84920111512&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.810

DO - https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.810

M3 - منشور من مؤتمر

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 810

EP - 825

BT - Leibniz International Proceedings in Informatics, LIPIcs

A2 - Jansen, Klaus

A2 - Rolim, Jose D. P.

A2 - Devanur, Nikhil R.

A2 - Moore, Cristopher

T2 - 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014

Y2 - 4 September 2014 through 6 September 2014

ER -