Theory of Computing
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Title : An O(√n) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow
Authors : Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd
Volume : 2
Number : 7
Pages : 137-146
URL : https://theoryofcomputing.org/articles/v002a007
Abstract
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We consider the maximization version of the edge-disjoint
path problem (EDP). In undirected graphs and directed
acyclic graphs, we obtain an O(sqrt n) upper bound on
the approximation ratio where n is the number of nodes
in the graph. We show this by establishing the upper bound
on the integrality gap of the natural relaxation based on
multicommodity flows. Our upper bound matches within a
constant factor a lower bound of Omega(sqrt n) that
is known for both undirected and directed acyclic graphs.
The best previous upper bounds on the integrality gaps were
O(min{n^{2/3}, sqrt{m}}) for undirected graphs and
O(min{sqrt{n log n}, sqrt{m}}) for directed acyclic graphs;
here m is the number of edges in the graph. These bounds
are also the best known approximation ratios for these problems.
Our bound also extends to the unsplittable flow problem (UFP)
when the maximum demand is at most the minimum capacity.