Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs
Combinatorics, Probability and Computing
Vertex percolation on expander graphs
European Journal of Combinatorics
Local resilience and hamiltonicity maker–breaker games in random regular graphs
Combinatorics, Probability and Computing
Approximation algorithms for edge-disjoint paths and unsplittable flow
Efficient Approximation and Online Algorithms
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Given a graph G=(V,E) with n vertices, m edges, and a family of $\kappa$ pairs of vertices in $V$, we are interested in finding for each pair (ai, bi) a path connecting ai to bi such that the set of $\kappa$ paths so found is edge disjoint. (For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed.)We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random graph Gn,m for all edge densities above the connectivity threshold. (The graph is chosen first; then an adversary chooses the pairs of endpoints.) Our results give the first tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.