87 research outputs found
Approximation algorithms for low-distortion embeddings into low-dimensional spaces
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 33-35).We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. We give an O([square root] n)-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. For the same problem, we give an exact algorithm, with running-time exponential in the distortion. We complement these results by showing that the problem is NP-hard to [alpha]-approximate, for some constant [alpha] > 1. For the two-dimensional case, we show a O([square root] n) upper bound for the distortion required to embed an n-point subset of the two-dimensional sphere, into the plane. We prove that this bound is asymptotically tight, by exhibiting n-point subsets such that any embedding into the plane has distortion [omega]([square root] n). These techniques yield a O(1)-approximation algorithm for the problem of embedding an n-point subset of the sphere into the plane.by Anastasios Sidiropoulos.S.M
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs
We present a near-optimal polynomial-time approximation algorithm for the
asymmetric traveling salesman problem for graphs of bounded orientable or
non-orientable genus. Our algorithm achieves an approximation factor of O(f(g))
on graphs with genus g, where f(n) is the best approximation factor achievable
in polynomial time on arbitrary n-vertex graphs. In particular, the
O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et
al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation
algorithm for genus-g graphs. Our result improves the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA
2011], which applies only to graphs with orientable genus g; ours is the first
approximation algorithm for graphs with bounded non-orientable genus.
Moreover, using recent progress on approximating the genus of a graph, our
O(log(g) / loglog(g))-approximation can be implemented even without an
embedding when the input graph has bounded degree. In contrast, the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a
genus-g embedding as part of the input.
Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on
graphs of genus g, with running time 2^O(g)*n^O(1)
Minimum d-dimensional arrangement with fixed points
In the Minimum -Dimensional Arrangement Problem (d-dimAP) we are given a
graph with edge weights, and the goal is to find a 1-1 map of the vertices into
(for some fixed dimension ) minimizing the total
weighted stretch of the edges. This problem arises in VLSI placement and chip
design.
Motivated by these applications, we consider a generalization of d-dimAP,
where the positions of some of the vertices (pins) is fixed and specified as
part of the input. We are asked to extend this partial map to a map of all the
vertices, again minimizing the weighted stretch of edges. This generalization,
which we refer to as d-dimAP+, arises naturally in these application domains
(since it can capture blocked-off parts of the board, or the requirement of
power-carrying pins to be in certain locations, etc.). Perhaps surprisingly,
very little is known about this problem from an approximation viewpoint.
For dimension , we obtain an -approximation
algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The
integrality gap for this LP is shown to be . We also show that
it is NP-hard to approximate 2-dimAP+ within a factor better than
\Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but
practically even more interesting) variant of 2-dimAP+, where the target space
is the grid , instead of
the entire integer lattice . For this problem, we obtain a -approximation using the same LP relaxation. We complement
this upper bound by showing an integrality gap of , and an
\Omega(k^{1/2-\eps})-inapproximability result.
Our results naturally extend to the case of arbitrary fixed target dimension
Non-positive curvature, and the planar embedding conjecture
The planar embedding conjecture asserts that any planar metric admits an
embedding into L_1 with constant distortion. This is a well-known open problem
with important algorithmic implications, and has received a lot of attention
over the past two decades. Despite significant efforts, it has been verified
only for some very restricted cases, while the general problem remains elusive.
In this paper we make progress towards resolving this conjecture. We show
that every planar metric of non-positive curvature admits a constant-distortion
embedding into L_1. This confirms the planar embedding conjecture for the case
of non-positively curved metrics
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