Maximum edge-disjoint paths in k-sums of graphs
Résumé
We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected
graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The
integrality gap is known to be Ω(√n) even for planar graphs [14] due to a simple topological obstruction
and a major focus, following earlier work [17], has been understanding the gap if some constant congestion
is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [23, 7]. In general graphs, recent
work has shown the gap to be polylog(n) [10, 11] with congestion 2. Moreover, the gap is log^Ω(c) n in
general graphs with congestion c for any constant c ≥ 1 [1].
It is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold.
It is easy to deduce that for given constant bounds on the approximation and congestion, the class of “nice”
graphs is minor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a
constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is
yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even
treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph
of bounded treewidth and is able to integrally route a constant fraction of the LP solution’s value. Note that
we do not incur any edge congestion. Previously this was not known even for series parallel graphs which
have treewidth 2. The algorithm is based on a more general argument that applies to k-sums of graphs in
some graph family, as long as the graph family has a constant factor, constant congestion bound. We then
use this to show that such bounds hold for the class of k-sums of bounded genus graphs.