1,889 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
Variants of Plane Diameter Completion
The {\sc Plane Diameter Completion} problem asks, given a plane graph and
a positive integer , if it is a spanning subgraph of a plane graph that
has diameter at most . We examine two variants of this problem where the
input comes with another parameter . In the first variant, called BPDC,
upper bounds the total number of edges to be added and in the second, called
BFPDC, upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in steps.Comment: Accepted in IPEC 201
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
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