2008 | OriginalPaper | Chapter
The Two-Edge Connectivity Survivable Network Problem in Planar Graphs
Authors : Glencora Borradaile, Philip Klein
Published in: Automata, Languages and Programming
Publisher: Springer Berlin Heidelberg
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Consider the following problem: given a graph with edge-weights and a subset
Q
of vertices, find a minimum-weight subgraph in which there are two edge-disjoint paths connecting every pair of vertices in
Q
. The problem is a failure-resilient analog of the Steiner tree problem, and arises in telecommunications applications. A more general formulation, also employed in telecommunications optimization, assigns a number (or
requirement
)
r
v
∈ {0,1,2} to each vertex
v
in the graph; for each pair
u
,
v
of vertices, the solution network is required to contain min{
r
u
,
r
v
} edge-disjoint
u
-to-
v
paths.
We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is SNP-hard in general graphs and NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is
O
(
n
log
n
).
Under the additional restriction that the requirements are in {0,2} for vertices on the boundary of a single face of a planar graph, we give a linear-time algorithm to find the optimal solution.