14,567 research outputs found
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
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
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
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