Data Correcting Algorithms in Combinatorial Optimization

This paper describes data correcting algorithms. It provides the theory behind the algorithms and presents the implementation details and computational experience with these algorithms on the asymmetric traveling salesperson problem, the problem of maximizing submodular functions, and the simple plant location problem.

Improved Algorithms for the Steiner Problem in Networks

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors

Class Notes in Discrete Mathematics, Operations Research, Statistics and Probability (Fourth Edition, v1)

Editorâ⠡‰ ¾Ã‚¢s Note: In graduate school, it became too cumbersome for me to look-up equations, theorems, proofs, and problem solutions from previous courses. I had three boxes full of notes and was going on my fourth. Due to the need to reference my notes periodically, the notes became more unorganized over time. Thatâ⠡‰ ¾Ã‚¢s when I decided to typeset them. I have been doing this for over a decade. Later in life, some colleagues asked if I could make these notes available to others (they were talking about themselves). I did. These notes can be downloaded for free from the web site http://www.repec.org/ and can be found in the Library of Congress. Note that the beginning of each chapter lists the professorâ⠡‰ ¾Ã‚¢s name and aï⠡½liation. Additionally, the course number, the date the course was taken, and the text book are given. The reader may also notice that I have made more use of the page space than in the previous editions of this manuscript. Hence, the book is shorter. If this causes the reader problems, then simply copy the proofs onto a blank sheet of paper â⠡¬â€� one line per algebraic manipulation. In this text, I put several algebraic manipulations on one line to save space

Algorithmic Approaches to the Steiner Problem in Networks

Das Steinerproblem in Netzwerken ist das Problem, in einem gewichteten Graphen eine gegebene Menge von Knoten kostenminimal zu verbinden. Es ist ein klassisches NP-schweres Problem und ein fundamentales Problem bei der Netzwerkoptimierung mit vielen praktischen Anwendungen. Wir nehmen dieses Problem mit verschiedenen Mitteln in Angriff: Relaxationen, die die Zulässigkeitsbedingungen lockern, um eine optimale Lösung annähern zu können; Heuristiken, um gute, aber nicht garantiert optimale Lösungen zu finden; und Reduktionen, um die Probleminstanzen zu vereinfachen, ohne eine optimale Lösung zu zerstören. In allen Fällen untersuchen und verbessern wir bestehende Methoden, stellen neue vor und evaluieren sie experimentell. Wir integrieren diese Bausteine in einen exakten Algorithmus, der den Stand der Algorithmik für die optimale Lösung dieses Problems darstellt. Viele der vorgestellten Methoden können auch für verwandte Probleme von Nutzen sein

Clustering-based Algorithms for Big Data Computations

In the age of big data, the amount of information that applications need to process often exceeds the computational capabilities of single machines. To cope with this deluge of data, new computational models have been defined. The MapReduce model allows the development of distributed algorithms targeted at large clusters, where each machine can only store a small fraction of the data. In the streaming model a single processor processes on-the-fly an incoming stream of data, using only limited memory. The specific characteristics of these models combined with the necessity of processing very large datasets rule out, in many cases, the adoption of known algorithmic strategies, prompting the development of new ones. In this context, clustering, the process of grouping together elements according to some proximity measure, is a valuable tool, which allows to build succinct summaries of the input data. In this thesis we develop novel algorithms for some fundamental problems, where clustering is a key ingredient to cope with very large instances or is itself the ultimate target. First, we consider the problem of approximating the diameter of an undirected graph, a fundamental metric in graph analytics, for which the known exact algorithms are too costly to use for very large inputs. We develop a MapReduce algorithm for this problem which, for the important class of graphs of bounded doubling dimension, features a polylogarithmic approximation guarantee, uses linear memory and executes in a number of parallel rounds that can be made sublinear in the input graph's diameter. To the best of our knowledge, ours is the first parallel algorithm with these guarantees. Our algorithm leverages a novel clustering primitive to extract a concise summary of the input graph on which to compute the diameter approximation. We complement our theoretical analysis with an extensive experimental evaluation, finding that our algorithm features an approximation quality significantly better than the theoretical upper bound and high scalability. Next, we consider the problem of clustering uncertain graphs, that is, graphs where each edge has a probability of existence, specified as part of the input. These graphs, whose applications range from biology to privacy in social networks, have an exponential number of possible deterministic realizations, which impose a big-data perspective. We develop the first algorithms for clustering uncertain graphs with provable approximation guarantees which aim at maximizing the probability that nodes be connected to the centers of their assigned clusters. A preliminary suite of experiments, provides evidence that the quality of the clusterings returned by our algorithms compare very favorably with respect to previous approaches with no theoretical guarantees. Finally, we deal with the problem of diversity maximization, which is a fundamental primitive in big data analytics: given a set of points in a metric space we are asked to provide a small subset maximizing some notion of diversity. We provide efficient streaming and MapReduce algorithms with approximation guarantees that can be made arbitrarily close to the ones of the best sequential algorithms available. The algorithms crucially rely on the use of a k-center clustering primitive to extract a succinct summary of the data and their analysis is expressed in terms of the doubling dimension of the input point set. Moreover, unlike previously known algorithms, ours feature an interesting tradeoff between approximation quality and memory requirements. Our theoretical findings are supported by the first experimental analysis of diversity maximization algorithms in streaming and MapReduce, which highlights the tradeoffs of our algorithms on both real-world and synthetic datasets. Moreover, our algorithms exhibit good scalability, and a significantly better performance than the approaches proposed in previous works

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum