Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a digraph D=(V,A) and a set of $\kappa$ pairs of vertices in V,  are interested in finding, for each pair (xi, yi), a directed path connecting xi to yi such that the set of $\kappa$ paths so found is arc-disjoint. For arbitrary graphs the problem is ${\cal NP}$-complete, even for $\kappa=2$. #R##N# present a polynomial time randomized algorithm for finding arc-disjoint paths in an r-regular expander digraph D.  show that if D has sufficiently strong expansion properties and the degree r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined. This is within a constant factor of best possible.
['Tom Bohman', 'Alan M. Frieze']
I am writing a research paper on a idea Arc-Disjoint Paths in Expander Digraphs
[{"from": "human", "value": "I am writing a research paper on a idea Arc-Disjoint Paths in Expander Digraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a digraph D=(V,A) and a set of $\\kappa$ pairs of vertices in V,  are interested in finding, for each pair (xi, yi), a directed path connecting xi to yi such that the set of $\\kappa$ paths so found is arc-disjoint. For arbitrary graphs the problem is ${\\cal NP}$-complete, even for $\\kappa=2$. #R##N# present a polynomial time randomized algorithm for finding arc-disjoint paths in an r-regular expander digraph D.  show that if D has sufficiently strong expansion properties and the degree r is sufficiently large, then all sets of $\\kappa=\\Omega(n/\\log n)$ pairs of vertices can be joined. This is within a constant factor of best possible."}]