9,773 research outputs found
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
Two-Level Rectilinear Steiner Trees
Given a set of terminals in the plane and a partition of into
subsets , a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree connecting the terminals in each set
() and a top-level tree connecting the trees . The goal is to minimize the total length of all trees. This problem
arises naturally in the design of low-power physical implementations of parity
functions on a computer chip.
For bounded we present a polynomial time approximation scheme (PTAS) that
is based on Arora's PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension. For the general case we propose an algorithm
that predetermines a connection point for each and
().
Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each and independently.
This gives us a -factor approximation with a running time of
suitable for fast practical computations. The
approximation factor reduces to by applying Arora's approximation scheme
in the plane
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
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