2 research outputs found
An FPT Algorithm Beating 2-Approximation for -Cut
In the -Cut problem, we are given an edge-weighted graph and an
integer , and have to remove a set of edges with minimum total weight so
that has at least connected components. Prior work on this problem
gives, for all , a -approximation algorithm for -cut
that runs in time . Hence to get a -approximation
algorithm for some absolute constant , the best runtime using
prior techniques is . Moreover, it was recently shown that
getting a -approximation for general is NP-hard,
assuming the Small Set Expansion Hypothesis.
If we use the size of the cut as the parameter, an FPT algorithm to find the
exact -Cut is known, but solving the -Cut problem exactly is -hard
if we parameterize only by the natural parameter of . An immediate question
is: \emph{can we approximate -Cut better in FPT-time, using as the
parameter?}
We answer this question positively. We show that for some absolute constant
, there exists a -approximation algorithm
that runs in time . This is the first FPT
algorithm that is parameterized only by and strictly improves the
-approximation.Comment: 26 pages, 4 figures, to appear in SODA '1
Losing Treewidth by Separating Subsets
We study the problem of deleting the smallest set of vertices (resp.
edges) from a given graph such that the induced subgraph (resp. subgraph)
belongs to some class . We consider the case where
graphs in have treewidth bounded by , and give a general
framework to obtain approximation algorithms for both vertex and edge-deletion
settings from approximation algorithms for certain natural graph partitioning
problems called -Subset Vertex Separator and -Subset Edge Separator,
respectively.
For the vertex deletion setting, our framework combined with the current best
result for -Subset Vertex Separator, yields a significant improvement in the
approximation ratios for basic problems such as -Treewidth Vertex Deletion
and Planar- Vertex Deletion. Our algorithms are simpler than previous works
and give the first uniform approximation algorithms under the natural
parameterization.
For the edge deletion setting, we give improved approximation algorithms for
-Subset Edge Separator combining ideas from LP relaxations and important
separators. We present their applications in bounded-degree graphs, and also
give an APX-hardness result for the edge deletion problems.Comment: 30 pages, 1 figure, to appear in SODA 1