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Iterative Rounding Approximation Algorithms in Network DesignShea, Marcus 05 1900 (has links)
Iterative rounding has been an increasingly popular approach to solving network design optimization problems ever since Jain introduced the concept in his revolutionary 2-approximation for the Survivable Network Design Problem (SNDP). This paper looks at several important iterative rounding approximation algorithms and makes improvements to some of their proofs. We generalize a matrix restatement of Nagarajan et al.'s token argument, which we can use to simplify the proofs of Jain's 2-approximation for SNDP and Fleischer et al.'s 2-approximation for the Element Connectivity (ELC) problem. Lau et al. show how one can construct a (2,2B + 3)-approximation for the degree bounded ELC problem, and this thesis provides the proof. We provide some structural results for basic feasible solutions of the Prize-Collecting Steiner Tree problem, and introduce a new problem that arises, which we call the Prize-Collecting Generalized Steiner Tree problem.
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Building Networks in the Face of UncertaintyGupta, Shubham January 2011 (has links)
The subject of this thesis is to study approximation algorithms for some network design problems in face of uncertainty. We consider two widely studied models of handling uncertainties - Robust Optimization and Stochastic Optimization. We study a robust version of the well studied Uncapacitated Facility Location Problem (UFLP). In this version, once the set of facilities to be opened is decided, an adversary may close at most β facilities. The clients must then be assigned to the remaining open facilities. The performance of a solution is measured by the worst possible set of facilities that the adversary may close. We introduce a novel LP for the problem, and provide an LP rounding algorithm when all facilities have same opening costs. We also study the 2-stage Stochastic version of the Steiner Tree Problem. In this version, the set of terminals to be covered is not known in advance. Instead, a probability distribution over the possible sets of terminals is known. One is allowed to build a partial solution in the first stage a low cost, and when the exact scenario to be covered becomes known in the second stage, one is allowed to extend the solution by building a recourse network, albeit at higher cost. The aim is to construct a solution of low cost in expectation. We provide an LP rounding algorithm for this problem that beats the current best known LP rounding based approximation algorithm.
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Algorithmic aspects of connectivity, allocation and design problemsChakrabarty, Deeparnab 23 May 2008 (has links)
Most combinatorial optimization problems are
NP -hard, which imply that under well- believed complexity assumptions, there exist no polynomial time
algorithms to solve them. To cope with the NP-hardness, approximation algorithms which return solutions close to
the optimal, have become a rich field of study. One successful method for designing approx-
imation algorithms has been to model the optimization problem as an integer program and
then using its polynomial time solvable linear programming relaxation for the design and
analysis of such algorithms. Such a technique is called the LP-based technique.
In this thesis, we study the algorithmic aspects of three classes of combinatorial optimization problems
using LP-based techniques as our main tool.
Connectivity Problems:
We study the Steiner tree problem and devise new linear pro-
gramming relaxations for the problem. We show an equivalence of our relaxation with the
well studied bidirected cut relaxation for the Steiner tree problem. Furthermore, for a class
of graphs called quasi-bipartite graphs, we improve the best known upper bound on the
integrality gap from 3/2 to 4/3. Algorithmically, we obtain fast and simple approximation
algorithms for the Steiner tree problem on quasi-bipartite graphs.
Allocation Problems:
We study the budgeted al location problem of allocating a set of
indivisible items to a set of agents who bid on it but possess a hard budget constraint more
than which they are unwilling to pay. This problem is a special case of submodular welfare
maximization. We use a natural LP relaxation for the problem and improve the best known
approximation factor for the problem from ~ 0.632 to 3/4. We also improve the inapprox-
imability factor of the problem to 15/16 and use our techniques to show inapproximability
results for many other allocation problems.
We also study online allocation problems where the set of items are unknown and appear one at a time.
Under some necessary assumptions we provide online algorithms for
many problems which attain the (almost) optimal competitive ratio. Both these works have
applications in the area of budgeted auctions, the most famous of which are the sponsored
search auctions hosted by search engines on the Internet.
Design Problems:
We formally define and study design problems which asks how the
weights of an input instance can be designed, so that the minimum (or maximum) of
a certain function of the input can be maximized (respectively, minimized). We show
if the function can be approximated to any factor $alpha$, then the optimum design can be
approximated to the same factor.
We also show that (max-min) design problems are dual to packing problems. We use
the framework developed by our study of design problems to obtain results about fraction-
ally packing Steiner trees in a "black-box" fashion. Finally, we study integral packing of
spanning trees and provide an alternate proof of a theorem of Nash-Williams and Tutte
about packing spanning trees.
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Algorithms for budgeted auctions and multi-agent covering problemsGoel, Gagan 07 July 2009 (has links)
In this thesis, we do an algorithmic study of
optimization problems in budgeted auctions, and some well known
covering problems in the multi-agent setting. We give new results
for the design of approximation algorithms, online algorithms and
hardness of approximation for these problems. Along the way we give
new insights for many other related problems.
Budgeted Auction. We study the following allocation problem which
arises in budgeted auctions (such as advertisement auctions run by
Google, Microsoft, Yahoo! etc.) : Given a set of m indivisible
items and n agents; agent i is willing to pay b[subscript ij] for item
j and has an overall budget of B[subscript i] (i.e. the maximum total
amount he is willing to pay). The goal is to allocate items to the
agents so as to maximize the total revenue obtained.
We study the computation complexity of the above allocation problem,
and give improved results for the approximation and the hardness of
approximation. We also study the above allocation problem in an
online setting. Online version of the problem has motivation in the
sponsored search auctions which are run by search engines. Lastly,
we propose a new bidding language for the budgeted auctions:
decreasing bid curves with budget constraints. We make a case for
why this language is better both for the sellers and for the buyers.
Multi-agent Covering Problems. To motivate this class of problems,
consider the network design problem of constructing a spanning tree
of a graph, assuming there are many agents willing to construct
different parts of the tree. The cost of each agent for constructing
a particular set of edges could be a complex function. For instance,
some agents might provide discounts depending on how many edges they
construct. The algorithmic question that one would be interested in
is: Can we find a spanning tree of minimum cost in polynomial time
in these complex settings? Note that such an algorithm will have to
find a spanning tree, and partition its edges among the agents.
Above are the type of questions that we are trying to answer for
various combinatorial problems. We look at the case when the agents'
cost functions are submodular. These functions form a rich class and
capture the natural properties of economies of scale or the law of
diminishing returns.We study the following fundamental problems in
this setting- Vertex Cover, Spanning Tree, Perfect Matching,
Reverse Auctions. We look at both the single agent and the
multi-agent case, and study the approximability of each of these
problems.
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Sink free orientations in a graphSivanathan, Gowrishankar. January 2009 (has links)
Thesis (M.S.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Computer Science, 2009. / Includes bibliographical references.
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Geometric optimization problems on orthogonal polygons: hardness results and approximation algorithmsMehrabidavoodabadi, Saeed 22 December 2015 (has links)
In this thesis, we design and develop new approximation algorithms and complexity results for three guarding and partitioning problems on orthogonal polygons; namely, guarding orthogonal polygons using sliding cameras, partitioning orthogonal polygons so as to minimize the stabbing number and guarding orthogonal terrains using vertex guards.
We first study a variant of the well-known art gallery problem in which sliding cameras are used to guard the polygon. We consider two versions of this problem: the Minimum- Cardinality Sliding Cameras (MCSC) problem in which we want to guard P with the minimum number of sliding cameras, and the Minimum-Length Sliding Cameras (MLSC) problem in which the goal is to compute a set S of sliding cameras for guarding P so as to minimize the total length of trajectories along which the cameras in S travel. We answer questions posed by Katz and Morgenstern (2011) by presenting the following results: (i) the MLSC problem is polynomially tractable even for orthogonal polygons with holes, (ii) the MCSC problem is NP-complete when P is allowed to have holes, and (iii) an O(n)-time exact algorithm for the MCSC problem on monotone polygons.
We then study a conforming variant of the problem of computing a partition of an orthogonal polygon P into rectangles whose stabbing number is minimum over all such partitions of P. The stabbing number of such a partition is the maximum number of rectangles intersected by any orthogonal line segment inside the polygon. In this thesis, we first give an O(n log n)-time algorithm that solves this problem exactly on histograms. We then show that the problem is NP-hard for orthogonal polygons with holes, providing the first hardness result for this problem. To complement the NP-hardness result, we give a 2-approximation algorithm for the problem on both polygons with and without holes.
Finally, we study a variant of the terrain guarding problem on orthogonal terrains in which the objective is to guard the vertices of an orthogonal terrain with the minimum number of vertex guards. We give a linear-time algorithm for this problem under a directed visibility constraint. / February 2016
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Approximation algorithms for facility location problems and other supply chain problems / Algoritmos de aproximação para problemas de alocação de instalações e outros problemas de cadeia de fornecimentoPedrosa, Lehilton Lelis Chaves, 1985- 07 April 2014 (has links)
Orientadores: Flávio Keidi Miyazawa, Maxim Sviridenko / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-25T09:17:37Z (GMT). No. of bitstreams: 1
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Previous issue date: 2014 / Resumo: O resumo poderá ser visualizado no texto completo da tese digital / Abstract: The abstract is available with the full electronic document / Doutorado / Ciência da Computação / Doutor em Ciência da Computação
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Uma ferramenta de auditoria para algoritmos de rearranjo de genomas / An audit tool for genome rearrangement algorithmsGalvão, Gustavo Rodrigues, 1988- 21 August 2018 (has links)
Orientador: Zanoni Dias / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-21T23:02:27Z (GMT). No. of bitstreams: 1
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Previous issue date: 2012 / Resumo: Ao longo da evolução, mutações globais podem alterar a ordem dos genes de um genoma. Tais mutações são chamadas de eventos de rearranjo. Em Rearranjo de Genomas, estimamos a distância evolutiva entre dois genomas calculando-se a distância de rearranjo entre eles, que é o tamanho da menor sequência de eventos de rearranjo que transforma um genoma no outro. Representando genomas como permutações, nas quais os genes aparecem como elemento, à distância de rearranjo pode ser obtido resolvendo-se o problema combinatório de ordenar uma permutação utilizando o menor número de eventos de rearranjo. Este problema, que é referido como Problema da Ordenação por Rearranjo, varia de acordo com os tipos de eventos de rearranjo considerados. Nesta dissertação, focamos nosso estudo em dois tipos de eventos: reversões e transposições. Variações do Problema da Ordenação por Rearranjo que consideram esses eventos têm se mostrado difíceis de ser resolvida otimamente, por isso a maior parte dos algoritmos propostos - os quais denominamos genericamente por algoritmos de rearranjo de genomas - são aproximados e é esperado que os próximos avanços ocorram nesse sentido. Em razão disso, desenvolvemos uma ferramenta que avalia as respostas desses algoritmos. Para ilustrar sua aplicação, nós a utilizamos para avaliar as respostas de 16 algoritmos de rearranjo de genomas aproximados relativos a 6 variações do Problema da Ordenação por Rearranjo. Além da ferramenta, este trabalho traz outras contribuições. Desenvolvemos um algoritmo exato para calcular distâncias de rearranjo que é mais eficiente em termos de uso de memória do que qualquer outro algoritmo que encontramos na literatura. Apresentamos conjecturas que dizem respeito à forma como as distâncias de rearranjo se distribuem. Validamos conjecturas referentes ao diâmetro, que é o maior valor alcançável pela distância de rearranjo entre uma permutação qualquer e a identidade considerando-se todas as permutações com o mesmo número de elementos. Apresentamos demonstrações formais para o fator de aproximação de alguns dos algoritmos avaliados. Por fim, mostramos que os fatores de aproximação de 7 dos 16 algoritmos avaliados não podem ser melhorados, o que contradiz algumas hipóteses levantadas na literatura, e conjecturamos que os fatores de aproximação de outros 6 algoritmos também não possam / Abstract: During evolution, global mutations may modify the gene order in a genome and such mutations are called rearrangement events. In Genome Rearrangements, we estimate the evolutionary distance between two genomes by computing the rearrangement distance between them, which is the length of the shortest sequence of rearrangement events that transforms one genome into the other. Representing genomes as permutations, in which genes appear as elements, the rearrangement distance can be obtained by solving the combinatorial problem of sorting a permutation using a minimum number of rearrangement events. This problem is referred to as Rearrangement Sorting Problem and varies accordingly to the types of rearrangement events considered. In this dissertation, we focus on two types of rearrangement events: reversals and transpositions. Variants of Rearrangement Sorting Problem involving these events have been shown to be difficult to solve optimally, therefore most of the proposed algorithms - which we denominate generically as genome rearrangement algorithms - are approximations, which have been the expected direction to follow. For this reason, we developed a tool that evaluates the results of these algorithms. To illustrate its application, we used it to evaluate the results of 16 genome rearrangement algorithms regarding 6 variants of Rearrangement Sorting Problem. Besides this tool, we developed an exact algorithm for computing rearrangement distances that is more efficient in terms of memory than any algorithm we have found in literature. Additionally, we presented conjectures on how the rearrangement distance are distributed and validated them regarding their diameter, which is the greatest value that the rearrangement distance between a permutation and the identity can reach considering all permutations with the same number of elements. Moreover, we presented formal proofs on the approximation ratio of some of the evaluated algorithms and showed that the approximation ratio of 7 out of the 16 evaluated algorithms cannot be improved, which contradicts some hypothesis raised in literature. Lastly, we conjectured that the approximation ratio of another 6 algorithms also cannot be improved / Mestrado / Ciência da Computação / Mestre em Ciência da Computação
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[en] IMPROVED APPROXIMATIONS FOR THE K-HOTLINK ASSIGNMENT PROBLEM AND FOR BINARY SEARCHING IN TREES / [pt] ALGORITMOS APROXIMATIVOS PARA O PROBLEMA DE ATRIBUIÇÃO DE HOTLINKS E PARA BUSCA BINÁRIA EM ÁRVORESMARCO SERPA MOLINARO 07 July 2008 (has links)
[pt] Neste trabalho, apresentamos algoritmos aproximativos para
dois problemas de otimização em árvores. Na primeira parte,
consideramos o Problema de Atribuição de k-Hotlinks. Seja G=
(V,E) um grafo direcionado acíclico representando um web
site, onde nós correspondem a páginas e arcos correspondem
a hyperlinks. Nesse contexto, hotlink são definidos como
atalhos (novos arcos) adicionados às páginas de G de modo a
reduzir o tempo gasto pelos usuários para alcançarem as
informações desejadas. Neste trabalho consideramos o
problema onde G é uma árvore enraizada e o objetivo é
minimizar o tempo médio gasto pelos usuários atribuindo no
máximo k hotlinks a cada nó. Para a versão mais estudada
desse problema onde no máximo um hotlink pode ser atribuído
a cada nó, provamos a existência de um FPTAS. Isso
representa uma significante melhora em relação ao algoritmo
com aproximação constante obtido recentemente em [Jacobs,
WADS 2007]. Além disso, desenvolvemos o primeiro algoritmo
com aproximação constante para a versão mais geral onde k
hotlinks podem ser atribuídos a cada nó. Na segunda parte
deste trabalho, consideramos o problema de computar
estratégias eficientes para realizar buscas em árvores.
Como uma generalização da tradicional busca binária em
listas ordenadas, suponha que se deseja encontrar um nó
específico (porém desconhecido) de uma árvore realizando
consultas em seus arcos, onde cada consulta indica a
extremidade do arco mais próxima ao nó desejado. Dada a
probabilidade de cada nó ser aquele procurado, o objetivo é
computar uma estratégia de busca que minimize o número
esperado de consultas. Aplicações práticas desse problema
incluem sincronização de file systems e testes de software.
Apresentamos um algoritmo linear que obtém a primeira
aproximação constante para esse problema. Isso representa
uma melhora significativa em relação à O(log n)-aproximação
anterior. / [en] Here we present a study on two optimization problems in
trees: the k-
Hotlink Assignment Problem and the problem of Binary
Searching in Trees.
As a result, we obtain improved approximation algorithms
for both problem.
The k-Hotlink Assignment Problem can be defined as follows.
Let G =
(V,E) be a directed acyclic graph representing a web site,
where nodes
correspond to pages and arcs to hyperlinks. In this
context, hotlinks are
defined as shortcuts (new arcs) added to web pages of G in
order to reduce
the time spent by users to reach their desired information.
Here we consider
the problem where G is a rooted directed tree and the goal
is minimizing the
expected time spent by users by assigning at most k
hotlinks to each node.
For the most studied version of this problem where at most
one hotlink
can be assigned from each node, we prove the existence of
an FPTAS,
improving upon the constant factor algorithm recently
obtained in [Jacobs,
WADS 2007]. In addition, we develop the first constant
factor approximation
algorithm for the most general version where k hotlinks can
be assigned from
each node.
In the second part of this work, we consider the problem of
computing efficient
strategies for searching in trees. As a generalization of
the classical
binary search for ordered lists, suppose one wishes to find
a (unknown) specific
node of a tree by asking queries to its arcs, where each
query indicates
the endpoint closer to the desired node. Given the
likelihood of each node
being the one searched, the objective is to compute a
search strategy that
minimizes the expected number of queries. Practical
applications of this
problem include file system synchronization and software
testing. Here we
present a linear time algorithm which is the first constant
factor approximation
for this problem. This represents a significant improvement
over
previous O(log n)-approximation.
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Energie, coopération méta-heuristiques et logique floue pour l'optimisation difficile / Energy, Cooperation Meta-heuristics and Fuzzy Logic for NP-hard OptimizationAutuori, Julien 05 December 2014 (has links)
Au cours de cette thèse, l'exploration de l'espace de solutions par des métaheuristiques est abordée. Les métaheuristiques sont des méthodes d'optimisation utilisées pour résoudre des problèmes NP-difficile. Elles explorent aléatoirement l'espace de recherche pour trouver les meilleures solutions. Dans un premier temps, l'ensemble des solutions est modélisé par un espace unidimensionnel par une Méthode de Conversion de l'Espace de recherche (MCE). Des métriques sont proposées pour évaluer l'exploration de l'espace de recherche par une métaheuristique en identifiant les zones explorées et inexplorées. Ces métriques sont utilisées pour orienter l'exploration de l'espace de recherche d'une méthode d'optimisation.La convergence est améliorée en accentuant le recherche dans les zones explorées. Pour sortir des minimums locaux, l'exploration est diversifiée en la dirigeant vers les zones inexplorées. En associant l'exploration du voisinage des solutions et ces métriques cartographiques, il est possible d'améliorer les performances des métaheuristiques. Plusieurs algorithmes mono-objectifs et multiobjectifs sont implémentés en version classique, hybridé par la recherche locale et par la MCE. Le Flexible Job Shop Problem (FJSP) est utilisé comme problème de référence. Les expérimentations avec les algorithmes hybridés montrent une amélioration des performances / In this thesis, the solution space exploration by the metaheuristic is developed. The metaheuristics optimization methods are used to solve NP-hard problems. They explore randomly the search space to look for the best solutions. In a first step, the solution set is modeled by a one-dimensional space by a Mapping Method (MaM). Metrics are proposed to evaluate the search space exploration by a metaheuristic, identifying the explored and unexplored zones. These metrics are used to guide the search space exploration of an optimization method. The convergence is improved by emphasizing the research in the zones explored. To get out local minima, the exploration is diversified by pointing it towards the unexplored zones. Combining the neighbour discovery of the solutions and these mapping metrics, it is possible to improve the performance of metaheuristics. Several single-objective and multi-objective algorithms are implemented in the classic version, hybridized with local search and MaM. The Flexible Job Shop Problem (FJSP) is used as a reference problem. The experimentations with hybridized algorithms show performance improved
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