40,516 research outputs found
Searching for Realizations of Finite Metric Spaces in Tight Spans
An important problem that commonly arises in areas such as internet
traffic-flow analysis, phylogenetics and electrical circuit design, is to find
a representation of any given metric on a finite set by an edge-weighted
graph, such that the total edge length of the graph is minimum over all such
graphs. Such a graph is called an optimal realization and finding such
realizations is known to be NP-hard. Recently Varone presented a heuristic
greedy algorithm for computing optimal realizations. Here we present an
alternative heuristic that exploits the relationship between realizations of
the metric and its so-called tight span . The tight span is a
canonical polytopal complex that can be associated to , and our approach
explores parts of for realizations in a way that is similar to the
classical simplex algorithm. We also provide computational results illustrating
the performance of our approach for different types of metrics, including
-distances and two-decomposable metrics for which it is provably possible
to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure
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
The Quadratic Cycle Cover Problem: special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of
node-disjoint cycles visiting all the nodes such that the total sum of
interaction costs between consecutive arcs is minimized. In this paper we study
the linearization problem for the quadratic cycle cover problem and related
lower bounds.
In particular, we derive various sufficient conditions for the quadratic cost
matrix to be linearizable, and use these conditions to compute bounds. We also
show how to use a sufficient condition for linearizability within an iterative
bounding procedure. In each step, our algorithm computes the best equivalent
representation of the quadratic cost matrix and its optimal linearizable matrix
with respect to the given sufficient condition for linearizability. Further, we
show that the classical Gilmore-Lawler type bound belongs to the family of
linearization based bounds, and therefore apply the above mentioned iterative
reformulation technique. We also prove that the linearization vectors resulting
from this iterative approach satisfy the constant value property.
The best among here introduced bounds outperform existing lower bounds when
taking both quality and efficiency into account
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