8,535 research outputs found
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
Coplanar constant mean curvature surfaces
We consider constant mean curvature surfaces of finite topology, properly
embedded in three-space in the sense of Alexandrov. Such surfaces with three
ends and genus zero were constructed and completely classified by the authors
in arXiv:math.DG/0102183. Here we extend the arguments to the case of an
arbitrary number of ends, under the assumption that the asymptotic axes of the
ends lie in a common plane: we construct and classify the entire family of
these genus-zero coplanar constant mean curvature surfaces.Comment: 35 pages, 10 figures; minor revisions including one new figure; to
appear in Comm. Anal. Geo
Heuristics with Performance Guarantees for the Minimum Number of Matches Problem in Heat Recovery Network Design
Heat exchanger network synthesis exploits excess heat by integrating process
hot and cold streams and improves energy efficiency by reducing utility usage.
Determining provably good solutions to the minimum number of matches is a
bottleneck of designing a heat recovery network using the sequential method.
This subproblem is an NP-hard mixed-integer linear program exhibiting
combinatorial explosion in the possible hot and cold stream configurations. We
explore this challenging optimization problem from a graph theoretic
perspective and correlate it with other special optimization problems such as
cost flow network and packing problems. In the case of a single temperature
interval, we develop a new optimization formulation without problematic big-M
parameters. We develop heuristic methods with performance guarantees using
three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy
packing. Numerical results from a collection of 51 instances substantiate the
strength of the methods
Solution of the Kirchhoff-Plateau problem
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in
which a flexible filament in the form of a closed loop is spanned by a liquid
film, with the filament being modeled as a Kirchhoff rod and the action of the
spanning surface being solely due to surface tension. We establish the
existence of an equilibrium shape that minimizes the total energy of the system
under the physical constraint of non-interpenetration of matter, but allowing
for points on the surface of the bounding loop to come into contact. In our
treatment, the bounding loop retains a finite cross-sectional thickness and a
nonvanishing volume, while the liquid film is represented by a set with finite
two-dimensional Hausdorff measure. Moreover, the region where the liquid film
touches the surface of the bounding loop is not prescribed a priori. Our
mathematical results substantiate the physical relevance of the chosen model.
Indeed, no matter how strong is the competition between surface tension and the
elastic response of the filament, the system is always able to adjust to
achieve a configuration that complies with the physical constraints encountered
in experiments
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