145 research outputs found
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
Fast Fencing
We consider very natural "fence enclosure" problems studied by Capoyleas,
Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a
set of points in the plane, we aim at finding a set of closed curves
such that (1) each point is enclosed by a curve and (2) the total length of the
curves is minimized. We consider two main variants. In the first variant, we
pay a unit cost per curve in addition to the total length of the curves. An
equivalent formulation of this version is that we have to enclose unit
disks, paying only the total length of the enclosing curves. In the other
variant, we are allowed to use at most closed curves and pay no cost per
curve.
For the variant with at most closed curves, we present an algorithm that
is polynomial in both and . For the variant with unit cost per curve, or
unit disks, we present a near-linear time algorithm.
Capoyleas, Rote, and Woeginger solved the problem with at most curves in
time. Arkin, Khuller, and Mitchell used this to solve the unit cost
per curve version in exponential time. At the time, they conjectured that the
problem with curves is NP-hard for general . Our polynomial time
algorithm refutes this unless P equals NP
A general branch-and-bound framework for continuous global multiobjective optimization
Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples
A topological solution to object segmentation and tracking
The world is composed of objects, the ground, and the sky. Visual perception
of objects requires solving two fundamental challenges: segmenting visual input
into discrete units, and tracking identities of these units despite appearance
changes due to object deformation, changing perspective, and dynamic occlusion.
Current computer vision approaches to segmentation and tracking that approach
human performance all require learning, raising the question: can objects be
segmented and tracked without learning? Here, we show that the mathematical
structure of light rays reflected from environment surfaces yields a natural
representation of persistent surfaces, and this surface representation provides
a solution to both the segmentation and tracking problems. We describe how to
generate this surface representation from continuous visual input, and
demonstrate that our approach can segment and invariantly track objects in
cluttered synthetic video despite severe appearance changes, without requiring
learning.Comment: 21 pages, 6 main figures, 3 supplemental figures, and supplementary
material containing mathematical proof
Application of general semi-infinite Programming to Lapidary Cutting Problems
We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented
Lagrangian duality in convex optimization.
Li, Xing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 76-80).Abstract also in Chinese.Introduction --- p.4Chapter 1 --- Preliminary --- p.6Chapter 1.1 --- Notations --- p.6Chapter 1.2 --- On Properties of Epigraphs --- p.10Chapter 1.3 --- Subdifferential Calculus --- p.14Chapter 1.4 --- Conical Approximations --- p.16Chapter 2 --- Duality in the Cone-convex System --- p.20Chapter 2.1 --- Introduction --- p.20Chapter 2.2 --- Various of Constraint Qualifications --- p.28Chapter 2.2.1 --- Slater´ةs Condition Revisited --- p.28Chapter 2.2.2 --- The Closed Cone Constrained Qualification --- p.31Chapter 2.2.3 --- The Basic Constraint Qualification --- p.38Chapter 2.3 --- Lagrange Multiplier and the Geometric Multiplier --- p.45Chapter 3 --- Stable Lagrangian Duality --- p.48Chapter 3.1 --- Introduction --- p.48Chapter 3.2 --- Stable Farkas Lemma --- p.48Chapter 3.3 --- Stable Duality --- p.57Chapter 4 --- Sequential Lagrange Multiplier Conditions --- p.63Chapter 4.1 --- Introduction --- p.63Chapter 4.2 --- The Sequential Lagrange Multiplier --- p.64Chapter 4.3 --- Application in Semi-Infinite Programs --- p.71Bibliography --- p.76List of Symbols --- p.8
On disconnected cuts and separators
Abstract. For a connected graph G = (V, E), a subset U ⊆ V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. A natural condition is to impose that for any u ∈ U the subgraph induced by (V \U ) ∪ {u} is connected. In that case U is called a minimal disconnected cut. We show that the problem of testing whether a graph has a minimal disconnected cut is NP-complete. We also show that the problem of testing whether a graph has a disconnected cut separating two specified vertices s and t is NP-complete
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