Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D

Algorithms and hardness results for the jump number problem, the joint replenishment problem, and the optimal clustering of frequency-constrained maintenance jobs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 107-110).In the first part of this thesis we present a new, geometric interpretation of the jump number problem on 2-dimensional 2-colorable (2D2C) partial order. We show that the jump number of a 2D2C poset is equivalent to the maximum cardinality of an independent set in a properly defined collection of rectangles in the plane. We then model the geometric problem as a linear program. Even though the underlying polytope may not be integral, we show that one can always find an integral optimal solution. Inspired by this result and by previous work of A. Frank, T. Jordan and L. Vegh [13, 14, 15] on set-pairs, we derive an efficient combinatorial algorithm to find the maximum independent set and its dual, the minimum hitting set, in polynomial time. The combinatorial algorithm solves the jump number problem on convex posets (a subclass of 2D2C posets) significantly faster than current methods. If n is the number of nodes in the partial order, our algorithm runs in 0((n log n)2.5) time, while previous algorithms ran in at least 0(n9 ) time. In the second part, we present a novel connection between certain sequencing problems that involve the coordination of activities and the problem of factorizing integer numbers. We use this connection to derive hardness results for three different problems: -- The Joint Replenishment Problem with General Integer Policies. -- The Joint Replenishment Problem with Correction Factor. -- The Problem of Optimal Clustering of Frequency-Constrained Maintenance Jobs. Our hardness results do not follow from a standard type of reduction (e.g., we do not prove NP-hardness), and imply that no polynomial-time algorithm exists for the problems above, unless Integer Factorization is solvable in polynomial time..by Claudio Telha Cornejo.Ph.D

Problem liczby skoków w zbiorach częściowo uporządkowanych. Kombinatoryczne algorytmy aproksymacyjne, przeszukiwanie wyczerpujące i złożoność obliczeniowa

Głównym problemem rozprawy doktorskiej jest minimalizacja liczby skoków posetu. Problem skoków dla danego posetu P polega na znalezieniu rozszerzenia liniowego, które minimalizuje liczbę sąsiadujących par elementów, nieporównywalnych w P. NP-trudność tego problemu została najpierw wykazana przez Pulleyblanka [56], a nastepnie na posetach przedziałowych przez Mitas [52]. Proponujemy kilka nowych algorytmów dla tego problemu. Najwięcej uwagi poświęcamy posetom przedziałowym. Dla posetów przedziałowych w latach 90-tych zaproponowano trzy wielomianowe algorytmy aproksymacyjne o współczynniku 3/2. Głównym rezultatem rozprawy jest przełamanie tego współczynnika. Poprawiamy algorytm podany przez Mitas i otrzymujemy aproksymację ze współczynnikiem 1.484. Ponadto przedstawiamy algorytm genetyczny dla problemu skoków na posetach przedziałowych, a także szybki algorytm dokładny dla tej klasy. W przypadku ogólnym, prezentujemy adaptację algorytmu przeszukiwania z zakazami działającą w oparciu o półsilnie zachłanne rozszerzenia liniowe, sformułowane przez Sysłę. Podejmujemy również temat posetów dwuwymiarowych. W klasie dwuwymiarowych posetów przedziałowych otrzymujemy algorytm aproksymacyjny dla problemu skoków ze współczynnikiem 4/3. Praktyczna część pracy obejmuje eksperymentalną analizę wydajności algorytmów przybliżajacych liczbę skoków.The main problem considered in this thesis is to minimize the jump number of a poset. The jump number problem for a given poset P is to find a linear extension minimizing the number of adjacent pairs which are incomparable in P. NP-hardness of this problem was first established by Pulleyblank [56], and later for interval orders by Mitas [52]. In the thesis, some new algorithms for this problem are proposed. We focus mainly on interval orders. In the 1990’s, three polynomial-time approximation algorithms have been given for interval orders with approximation ratio of 3/2. The main result of this thesis is an improvement of this approximation ratio. We enhance the algorithm given by Mitas and we obtain a 1.484-approximation algorithm. Moreover, we present a genetic algorithm for the jump number problem on interval orders, and a fast exact algorithm for this class. In the general case, we present an adaptation of the tabu search technique, based on semi-strongly greedy linear extensions, defined by Sysło. We also undertake the jump number of two-dimensional orders. We obtain a 4/3-approximation algorithm for the class of two-dimensional interval orders. In addition, the thesis contains an experimental analysis of efficiency of algorithms to approximate the jump number

The Camp Rayner Site (EgNr-2) : archaeological investigations of a multi-component site in south-central Saskatchewan

The Camp Rayner site (EgNr-2) is a multicomponent site located approximately 135km south of Saskatoon, Saskatchewan and is situated along the northern shoreline of Lake Diefenbaker and the western shoreline of Hitchcock Bay. The Saskatchewan Archaeological Society conducted field school excavations at Camp Rayner between the years of 1987 and 1995 as part of a salvage/rescue program for reasons of potential heritage displacement and site destruction. In total, 53 1x1m2 units were opened and revealed 7 occupation levels that span the Terminal Late Paleoindian to the Late Precontact period. Two radiocarbon dates were obtained which corroborates with both the Terminal Late Paleoindian and Early Middle Period. Research included an analysis of the entire cultural assemblage to reconstruct the cultural sequence of the site. This site offers a unique opportunity to study a number of archaeological cultures on the Northern Plains. The presence of an in situ Terminal Late Paleoindian and Early Middle Period occupation with correlating radiocarbon dates are of considerable significance due to their rarity on the northern grasslands. The recovery of Sandy Creek points and other Late Middle Period projectile points are also regarded as especially significant due to an increase in cultural complexity during the Late Middle and Late Precontact periods. The Camp Rayner site is one of the most significant sites in Saskatchewan. Cultural material at this site represents the last 9,000 years of human occupation with in situ deposits spanning approximately 7,000 years ago. The continuous investigation and monitoring of the archaeological record recovered at this site is the key to maintaining these non-renewable resources. The information gathered from this research will supplement research on archaeological occupations of the Northern Plains and will initiate a resource management plan for future excavations and site preservation

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Sea Level Fluctuations

This thematic issue presents 11 scientific articles that are extremely useful for understanding the processes and phenomena of the interacting geospheres of the Earth. These processes have an important impact on the biosphere and many human activities. The results of scientific research presented in this book are fully united by the common theme "investigation of the fundamental foundations of the emergence, development, transformation, and interaction of hydroacoustic, hydrophysical and geophysical fields in the World Ocean." The book is recommended to a wide range of readers, as well as to specialists in the field of hydroacoustics, oceanology, and geophysics

Número acromático de gráficas gramíneas bipartitas

44 páginas. Maestría en Optimización.En este trabajo estudiamos diversas propiedades de las gráficas gramíneas bipartitas, enfocándonos en particular en las coloraciones completas y el número acromático de las mismas. En el capítulo 1, presentamos al lector los conceptos preliminares más importantes para el desarrollo de éste trabajo. En el capíutlo 2, introducimos una clasificación de las gramíneas bipartitas en varias familias, y presentamos varias propiedades relacionadas con la estructura de estas familias, en particular, mostramos dos resultados importantes: una caracterización de un grupo de gramíneas bipartitas en términos de acoplamientos y la relación que el mismo grupo guarda con la familia de torneos regulares. También exploramos el problema de reconocer gráficas gramíneas en estas familias y presentamos un programa entero y un algoritmo que resuelven el problema. En cuanto a problemas de coloración, en el capítulo 3, damos una cota superior, que es justa, para el número acromático de una familia de gramíneas bipartitas y clasificamos las coloraciones completas que alcanzan dicha cota. Estudiamos algunas de las coloraciones completas mencionadas y exhibimos condiciones necesarias y condiciones suficientes para la existencia de estas coloraciones. Así mismo, presentamos técnicas para obtener y extender coloraciones completas en las gráficas de interés

Real-time analysis of video signals

Many practical and experimental systems employing image processing techniques have been built by other workers for various applications. Most of these systems are computer-based and very few operate in a real time environment. The objective of this work is to build a microprocessor-based system for video image processing. The system is used in conjunction with an on-line TV camera and processing is carried out in real time. The enormous storage requirement of digitized TV signals and the real time constraint suggest that some simplification of the data must take place prior to any viable processing. Data reduction is attained through the representation of objects by their edges, an approach often adopted for feature extraction in pattern recognition systems. A new technique for edge detection by applying comparison criteria to differentials at adjacent pixels of the video image is developed and implemented as a preprocessing hardware unit. A circuit for the generation of the co-ordinates of edge points is constructed to free the processing computer of this task, allowing it more time for on-line analysis of video signals. Besides the edge detector and co-ordinate generator the hardware built consists of a microprocessor system based on a Texas Instruments T.US 9900 device, a first-in-first-out buffer store and interface circuitry to a TV camera and display devices. All hardware modules and their power supplies are assembled in one unit to provide a standalone instrument. The problem chosen for investigation is analysis of motion in a visual scene. Aspects of motion studied concern the tracking of moving objects with simple geometric shapes and description of their motion. More emphasis is paid to the analysis of human eye movements and measurement of its point-of-regard which has many practical applications in the fields of physiology and psychology. This study provides a basis for the design of a processing unit attached to an oculometer to replace bulky minicomputer-based eye motion analysis systems. Programs are written for storage, analysis and display of results in real time

Hyperspectral Image Unmixing Incorporating Adjacency Information

While the spectral information contained in hyperspectral images is rich, the spatial resolution of such images is in many cases very low. Many pixel spectra are mixtures of pure materials’ spectra and therefore need to be decomposed into their constituents. This work investigates new decomposition methods taking into account spectral, spatial and global 3D adjacency information. This allows for faster and more accurate decomposition results