91 research outputs found
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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Topology Network Optimization of Facility Planning and Design Problems
The research attempts to provide a graphical theory-based approach to solve the facility layout problem. Which has generally been approached using Quadratic Assignment Problem (QAP) in the past, an algebraic method. It is a very complex problem and is part of the NP-Hard optimization class, because of the nonlinear quadratic objective function and (0,1) binary variables. The research is divided into three phases which together provide an optimal facility layout, block plan solution consisting of MHS (material handling solution) projected onto the block plan. In phase one, we solve for the position of departments in a facility based on flow and utility factor (weight for location). The position of all the departments is identified on the vertices of MPG (maximal planar graph), which maximizes the possibility of flow. We use named MPG produced in literature, throughout the research. The grouping of the department is achieved through GMAFLAD, a QSP (quadratic set packing) based optimizer. In Phase 2, the dual for the MPGâs is solved consisting of department location as per phase 1, to generate Voronoi graphs. These graphs are then, expanded by an ingenious parameter optimization formulation to achieve area fitting for individual cases. Optimization modeling software, Lingo17.0 is used for solving the parameter optimization for generating coordinates of the block plan. The plotting of coordinates for the block plan graphics is done via Autodesk inventor 2019. In phase 3, the solution for MHS is achieved using an RSMT (Rectilinear Steiner minimal tree) graph approach. The Voronoi seed coordinates produced through phase 2 results are computed by GeoSteiner package to generated the RSMT graph for projection onto the block plan (Also, done by Inventor 2019). The graphical method employed in this research, itself has complex and NP-hard problem segments in it, which have been relaxed to polynomial time complexity by fragmenting into groups and solving them in sections. Solving for MPG & RSMT are a class of NP-Hard problem, which have been restricted to N=32 here. Finally, to validate the research and its methodology a real-life case study of a shipyard building for the data set of PDVSA, Venezuela is performed and verified
A complete design path for the layout of flexible macros
XIV+172hlm.;24c
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments
The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the systemâs components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades.
A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions.
The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated.
Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered
Algorithm engineering in geometric network planning and data mining
The geometric nature of computational problems provides a rich source of solution strategies as well as complicating obstacles. This thesis considers three problems in the context of geometric network planning, data mining and spherical geometry.
Geometric Network Planning:
In the d-dimensional Generalized Minimum Manhattan Network problem (d-GMMN) one is interested in finding a minimum cost rectilinear network N connecting a given set of n pairs of points in â^d such that each pair is connected in N via a shortest Manhattan path. The decision version of this optimization problem is known to be NP-hard. The best known upper bound is an O(log^{d+1} n) approximation for d>2 and an O(log n) approximation for 2-GMMN.
In this work we provide some more insight in, whether the problem admits constant factor approximations in polynomial time. We develop two new algorithms, a `scale-diversity aware' algorithm with an O(D) approximation guarantee for 2-GMMN. Here D is a measure for the different `scales' that appear in the input, D â O(log n) but potentially much smaller, depending on the problem instance. The other algorithm is based on a primal-dual scheme solving a more general, combinatorial problem - which we call Path Cover. On 2-GMMN it performs well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended to deal with obstacle avoiding requirements. We show that the Path Cover problem is at least as hard to approximate as the Hitting Set problem. Moreover, we show that solutions of the primal-dual algorithm are 4Ï^2 approximations, where Ï â€ n denotes the maximum overlap of a problem instance. This implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales and non-constant overlap in the construction.
Geometric Map Matching for Heterogeneous Data:
For a given sequence of location measurements, the goal of the geometric map matching is to compute a sequence of movements along edges of a spatially embedded graph which provides a `good explanation' for the measurements.
The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precision. Hence, formalizing what a `good explanation' is becomes quite difficult.
We propose a novel map matching approach, which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching, even in presence of highly variable data quality without parameter tuning.
Rational Points on the Unit Spheres:
Each non-zero point in â^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an Δ-approximation y â â^d for x, that is exactly on S^{d-1} and has low bit-size. We revise lower bounds on rational approximations and provide explicit spherical instances.
We prove that floating-point numbers can only provide trivial solutions to the sphere equation in â^2 and â^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/Δ^2 for any given Δ â (0, 1/8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation.
Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets, geo-referenced by latitude and longitude values.Die geometrische Gestalt von Berechnungsproblemen liefert vielfÀltige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphÀrischen Geometrie.
Geometrische Netzwerk Planung:
Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein gĂŒnstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus â^d mit einem kĂŒrzesten Manhattan Pfad verbindet. Es ist bekannt, dass die Entscheidungsvariante dieses Optimierungsproblems NP-hart ist. Die beste bekannte obere Schranke ist eine O(log^{d+1} n) Approximation fĂŒr d>2 und eine O(log n) Approximation fĂŒr 2-GMMN.
Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulĂ€sst. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `SkalendiversitĂ€t' und hat eine O(D) ApproximationsgĂŒte fĂŒr 2-GMMN. Hierbei ist D ein MaĂ fĂŒr die in Eingaben auftretende `Skalen'. D â O(log n), aber potentiell deutlichen kleiner fĂŒr manche Problem Instanzen. Der andere Algorithmus basiert auf einem Primal-Dual Schema zur Lösung eines allgemeineren, kombinatorischen Problems, welches wir Path Cover nennen. Die praktisch erzielten a posteriori ApproximationsgĂŒten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann fĂŒr Netzwerk Planungsprobleme mit Hindernis-Anforderungen angepasst werden. Wir zeigen, dass das Path Cover Problem mindestens so schwierig zu approximieren ist wie das Hitting Set Problem. DarĂŒber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4Ï^2 Approximationen sind, wobei Ï â€ n die maximale Ăberlappung einer Probleminstanz bezeichnet. Daher mĂŒssen potentielle Beweise, die konstante Approximationen fĂŒr 2-GMMN ausschlieĂen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Ăberlappung konstruieren.
Geometrisches Map Matching fĂŒr heterogene Daten:
FĂŒr eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines rĂ€umlich eingebetteten Graphen zu finden, welche eine `gute ErklĂ€rung' fĂŒr die Messungen ist.
Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene DatenqualitÀt aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute ErklÀrung' ist, wird dadurch schwer.
Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der DatenqualitĂ€t anpasst indem er sogenannte Dominance Decompositions berechnet. Obwohl unser Ansatz teurer im Rechenaufwand ist, zeigen unsere Experimente, dass qualitativ hochwertige Map Matching Ergebnisse auf hoch variabler DatenqualitĂ€t erzielbar sind ohne vorher Parameter kalibrieren zu mĂŒssen.
Rationale Punkte auf EinheitssphÀren:
Jeder, von Null verschiedene, Punkt in â^d identifiziert einen nĂ€chsten Punkt x auf der EinheitssphĂ€re S^{d-1}. Wir suchen eine Δ-Approximation y â â^d fĂŒr x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-GröĂe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphĂ€rische Instanzen.
Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur SphĂ€ren-Gleichung in â^2 und â^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/Δ^2 sind fĂŒr gegebene Δ â (0, 1/8], was ein bekanntes Resultat verbessert. DarĂŒber hinaus profitiert die Methode von Algorithmen fĂŒr simultane Diophantische Approximationen.
Unsere quell-offene Implementierung und die Experimente demonstrieren die PraktikabilitĂ€t unseres Ansatzes fĂŒr sehr groĂe, durch geometrische LĂ€ngen- und Breitengrade referenzierte, DatensĂ€tze
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