5,796 research outputs found
A face cover perspective to embeddings of planar graphs
It was conjectured by Gupta et al. [Combinatorica04] that every planar graph
can be embedded into with constant distortion. However, given an
-vertex weighted planar graph, the best upper bound on the distortion is
only , by Rao [SoCG99]. In this paper we study the case where
there is a set of terminals, and the goal is to embed only the terminals
into with low distortion. In a seminal paper, Okamura and Seymour
[J.Comb.Theory81] showed that if all the terminals lie on a single face, they
can be embedded isometrically into . The more general case, where the
set of terminals can be covered by faces, was studied by Lee and
Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the
art is an upper bound of by Krauthgamer, Lee and Rika
[SODA19]. Our contribution is a further improvement on the upper bound to
. Since every planar graph has at most faces, any
further improvement on this result, will be a major breakthrough, directly
improving upon Rao's long standing upper bound. Moreover, it is well known that
the flow-cut gap equals to the distortion of the best embedding into .
Therefore, our result provides a polynomial time -approximation to the sparsest cut problem on planar graphs, for the
case where all the demand pairs can be covered by faces
Two-sets cut-uncut on planar graphs
We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein,
one is given an undirected planar graph and two sets of vertices and
. The question is, what is the minimum number of edges to remove from ,
such that we separate all of from all of , while maintaining that every
vertex in , and respectively in , stays in the same connected component.
We show that this problem can be solved in time with a
one-sided error randomized algorithm. Our algorithm implies a polynomial-time
algorithm for the network diversion problem on planar graphs, which resolves an
open question from the literature. More generally, we show that Two-Sets
Cut-Uncut remains fixed-parameter tractable even when parameterized by the
number of faces in the plane graph covering the terminals , by
providing an algorithm of running time .Comment: 22 pages, 5 figure
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