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Introduction to Algorithms, second edition
 BOOK
, 2001
"... This part will get you started in thinking about designing and analyzing algorithms.
It is intended to be a gentle introduction to how we specify algorithms, some of the
design strategies we will use throughout this book, and many of the fundamental
ideas used in algorithm analysis. Later parts of t ..."
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Cited by 707 (3 self)
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This part will get you started in thinking about designing and analyzing algorithms.
It is intended to be a gentle introduction to how we specify algorithms, some of the
design strategies we will use throughout this book, and many of the fundamental
ideas used in algorithm analysis. Later parts of this book will build upon this base.
Chapter 1 is an overview of algorithms and their place in modern computing
systems. This chapter defines what an algorithm is and lists some examples. It also
makes a case that algorithms are a technology, just as are fast hardware, graphical
user interfaces, objectoriented systems, and networks.
In Chapter 2, we see our first algorithms, which solve the problem of sorting
a sequence of n numbers. They are written in a pseudocode which, although not
directly translatable to any conventional programming language, conveys the structure
of the algorithm clearly enough that a competent programmer can implement
it in the language of his choice. The sorting algorithms we examine are insertion
sort, which uses an incremental approach, and merge sort, which uses a recursive
technique known as “divide and conquer.” Although the time each requires increases
with the value of n, the rate of increase differs between the two algorithms.
We determine these running times in Chapter 2, and we develop a useful notation
to express them.
Chapter 3 precisely defines this notation, which we call asymptotic notation. It
starts by defining several asymptotic notations, which we use for bounding algorithm
running times from above and/or below. The rest of Chapter 3 is primarily a
presentation of mathematical notation. Its purpose is more to ensure that your use
of notation matches that in this book than to teach you new mathematical concepts.
Invitation to FixedParameter Algorithms
, 2002
"... Contents 1. Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Keep the Parameter Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries and Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..."
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Cited by 293 (73 self)
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Contents 1. Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Keep the Parameter Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries and Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Parameterized Complexitya Brief Overview . . . . . . . . . . . . . . 6 1.3.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Interpreting FixedParameter Tractability . . . . . . . . . . . 9 1.4 Vertex Cover  an Illustrative Example . . . . . . . . . . . . . . . . . 11 1.4.1 Parameterize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Specialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Generalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.4 Count or Enumerate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 238 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
Complexity and Approximation
, 1999
"... Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance ..."
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Cited by 176 (1 self)
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Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance I. How can we exploit the knowledge we have on the solution of instance I to compute a (approximate) solution of instance I ′ in an efficient way? This computation model is called reoptimization and is of practical interest in various circumstances. In this article we first discuss what kind of performance we can expect for specific classes of problems and then we present some classical optimization problems (i.e. Max Knapsack, Min Steiner Tree, Scheduling) in which this approach has been fruitfully applied. Subsequently, we address vehicle routing problems and we show how the reoptimization approach can be used to obtain good approximate solution in an efficient way for some of these problems. 1
Winner determination in combinatorial auction generalizations
, 2002
"... Combinatorial markets where bids can be submitted on bundles of items can be economically desirable coordination mechanisms in multiagent systems where the items exhibit complementarity and substitutability. There has been a surge of recent research on winner determination in combinatorial auctions. ..."
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Cited by 157 (23 self)
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Combinatorial markets where bids can be submitted on bundles of items can be economically desirable coordination mechanisms in multiagent systems where the items exhibit complementarity and substitutability. There has been a surge of recent research on winner determination in combinatorial auctions. In this paper we study a wider range of combinatorial market designs: auctions, reverse auctions, and exchanges, with one or multiple units of each item, with and without free disposal. We first theoretically characterize the complexity. The most interesting results are that reverse auctions with free disposal can be approximated, and in all of the cases without free disposal, even finding a feasible solution is ÆÈcomplete. We then ran experiments on known benchmarks as well as ones which we introduced, to study the complexity of the market variants in practice. Cases with free disposal tended to be easier than ones without. On many distributions, reverse auctions with free disposal were easier than auctions with free disposal— as the approximability would suggest—but interestingly, on one of the most realistic distributions they were harder. Singleunit exchanges were easy, but multiunit exchanges were extremely hard. 1
Computational Complexity  A Modern Approach
, 2009
"... Not to be reproduced or distributed without the authors ’ permissioniiTo our wives — Silvia and RavitivAbout this book Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these incl ..."
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Cited by 151 (2 self)
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Not to be reproduced or distributed without the authors ’ permissioniiTo our wives — Silvia and RavitivAbout this book Computational complexity theory has developed rapidly in the past three decades. The list of surprising and fundamental results proved since 1990 alone could fill a book: these include new probabilistic definitions of classical complexity classes (IP = PSPACE and the PCP Theorems) and their implications for the field of approximation algorithms; Shor’s algorithm to factor integers using a quantum computer; an understanding of why current approaches to the famous P versus NP will not be successful; a theory of derandomization and pseudorandomness based upon computational hardness; and beautiful constructions of pseudorandom objects such as extractors and expanders. This book aims to describe such recent achievements of complexity theory in the context of more classical results. It is intended to both serve as a textbook and as a reference for selfstudy. This means it must simultaneously cater to many audiences, and it is carefully designed with that goal. We assume essentially no computational background and very minimal mathematical background, which we review in Appendix A. We have also provided a web site for this book at
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 123 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
The communication requirements of efficient allocations and supporting prices
 Journal of Economic Theory
, 2006
"... We show that any communication finding a Pareto efficient allocation in a privateinformation economy must also discover supporting Lindahl prices. In particular, efficient allocation of L indivisible objects requires naming a price for each of the 2 L ¡1 bundles. Furthermore, exponential communicat ..."
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Cited by 113 (15 self)
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We show that any communication finding a Pareto efficient allocation in a privateinformation economy must also discover supporting Lindahl prices. In particular, efficient allocation of L indivisible objects requires naming a price for each of the 2 L ¡1 bundles. Furthermore, exponential communication in L is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the valuations are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the objects are homogeneous, arbitrarily good approximation is obtained using exponentially less communication than that needed for exact efficiency.