TY - GEN

T1 - Rolling backwards can move you forward

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

AU - Draganić, Nemanja

AU - Krivelevich, Michael

AU - Nenadov, Rajko

N1 - Publisher Copyright: Copyright © 2021 by SIAM

PY - 2021

Y1 - 2021

N2 - We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems: • For a graph H, we denote by Hq the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number R^k(Hq) satisfies R^k(Hq) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak (2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d1−ε), and let P be any collection of at most cnloglognd disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length l = O ( loglognd ). Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

AB - We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems: • For a graph H, we denote by Hq the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number R^k(Hq) satisfies R^k(Hq) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak (2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d1−ε), and let P be any collection of at most cnloglognd disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length l = O ( loglognd ). Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

UR - http://www.scopus.com/inward/record.url?scp=85102794273&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/1.9781611976465

DO - https://doi.org/10.1137/1.9781611976465

M3 - منشور من مؤتمر

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 123

EP - 134

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

Y2 - 10 January 2021 through 13 January 2021

ER -