Efficient Algorithms for Bipartite Matching Problems with Preferences PDF Download
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Author: Colin Sng
Publisher:
ISBN:
Category : Algorithms
Languages : en
Pages : 149
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Book Description
Author: Colin Sng
Publisher:
ISBN:
Category : Algorithms
Languages : en
Pages : 149
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Book Description
Author: Colin Sng
Publisher:
ISBN:
Category : Algorithms
Languages : en
Pages : 0
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Book Description
Author: David Manlove
Publisher: World Scientific
ISBN: 9814425265
Category : Computers
Languages : en
Pages : 524
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Book Description
Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists.In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley.This book describes the most important results in this area, providing a timely update to The Stable Marriage Problem: Structure and Algorithms (D Gusfield and R W Irving, MIT Press, 1989) in connection with stable matching problems, whilst also broadening the scope to include matching problems with preferences under a range of alternative optimality criteria.
Author: Mikhail Kapralov
Publisher:
ISBN:
Category :
Languages : en
Pages :
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The problem of finding maximum matchings in bipartite graphs is a classical problem in combinatorial optimization with a long algorithmic history. Graph sparsification is a more recent paradigm of replacing a graph with a smaller subgraph that preserves some useful properties of the original graph, perhaps approximately. Traditionally, sparsification has been used for obtaining faster algorithms for cut-based optimization problems. The contributions of this thesis are centered around new algorithms for bipartite matching problems, in which, surprisingly, graph sparsification plays a major role, and efficient algorithms for constructing sparsifiers in modern data models. In the first part of the thesis we develop sublinear time algorithms for finding perfect matchings in regular bipartite graphs. These graphs have been studied extensively in the context of expander constructions, and have several applications in combinatorial optimization. The problem of finding perfect matchings in regular bipartite graphs has seen almost 100 years of algorithmic history, with the first algorithm dating back to K\"onig in 1916 and an algorithm with runtime linear in the number of edges in the graph discovered in 2000. In this thesis we show that, even though traditionally the use of sparsification has been restricted to cut-based problems, in fact sparsification yields extremely efficient {\em sublinear time} algorithms for finding perfect matchings in regular bipartite graphs when the graph is given in adjacency array representation. Thus, our algorithms recover a perfect matching (with high probability) without looking the whole input. We present two approaches, one based on independent sampling and another on random walks, obtaining an algorithm that recovers a perfect matching in $O(n\log n)$ time, within $O(\log n)$ of output complexity, essentially closing the problem. In the second part of the thesis we study the streaming complexity of maximum bipartite matching. This problem is relevant to modern data models, where the algorithm is constrained in space and is only allowed few passes over the input. We are interested in determining the best tradeoff between the space usage and the quality of the solution obtained. We first study the problem in the single pass setting. A central object of our study is a new notion of sparsification relevant to matching problems: we define the notion of an $\e$-matching cover of a bipartite graph as a subgraph that approximately preserves sizes of matchings between every two subsets of vertices, which can be viewed as a 'sparsifier' for matching problems. We give an efficient construction of a sparse subgraph that we call a 'matching skeleton', which we show is a linear-size matching cover for a certain range of parameters (in fact, for $\e> 1/2$). We then show that our 'sparsifier' can be applied repeatedly while maintaining a non-trivial approximation ratio in the streaming model with vertex arrivals, obtaining the first $1-1/e$ deterministic one-pass streaming algorithm that uses linear space for this setting. Further, we show that this is in fact best possible: no algorithm can obtain a better than $1-1/e$ approximation in a single pass unless it uses significantly more than quasilinear space. This is a rather striking conclusion since a $1-1/e$ approximation can be obtained even in the more restrictive online model for this setting. Thus, we show that streaming algorithms can get no advantage over online algorithms for this problem unless they use substantially more than quasilinear space. Our impossibility results for approximating matchings in a single pass using small space exploit a surprising connection between the sparsifiers that we define and a family of graphs known as \rs graphs. In particular, we show that bounding the best possible size of $\e$-covers for general $\e$ is essentially equivalent to determining the optimal size of an $\e$-\rs graph. These graphs have received significant attention due to applications in PCP constructions, property testing and additive combinatorics, but determining their optimal size still remains a challenging open problem. Besides giving matching upper and lower bounds for single pass algorithms in the vertex arrival setting, we also consider the problem of approximating matchings in multiple passes. Here we give an algorithm that achieves a factor of $1-e^{-k}k^{k}/k!=1-\frac{1}{\sqrt{2\pi k}}+o(1/k)$ in $k$ passes, improving upon the previously best known approximation. In the third part of the thesis we consider the concept of {\em spectral sparsification} introduced by Spielman and Teng. Here, we uncover a connection between spectral sparsification and spanners, i.e. subgraphs that approximately preserve shortest path distances. This connection allows us to obtain a quasilinear time algorithm for constructing spectral sparsifiers using approximate distance oracles and entirely bypassing linear system solvers, which was previously the only known way of constructing spectral sparsifiers in quasilinear time. Finally, in the last part of the thesis we design an efficient implementation of cut-preserving sparsification in a streaming setting with edge deletions using only one pass over the data.
Author: J. Robert Kennedy
Publisher:
ISBN:
Category : Computer algorithms
Languages : en
Pages : 222
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Abstract: "The push-relabel method has been shown to be efficient for solving maximum flow and minimum cost flow problems in practice, and periodic global updates of dual variables have played an important role in the best implementations. Nevertheless, global updates had not been known to yield any theoretical improvement in running time. In this work, we study techniques for implementing push-relabel algorithms to solve bipartite matching and assignment problems. We show that global updates yield a theoretical improvement in the bipartite matching and assignment contexts, and we develop a suite of efficient cost-scaling push-relabel implementations to solve assignment problems. For bipartite matching, we show that a push-relabel algorithm using global updates runs in [formula] time (matching the best bound known) and performs worse by a factor of [square root of n] without the updates. We present a similar result for the assignment problem, for which an algorithm that assumes integer costs in the range [-C ..., C] runs in time O([square root of nm] log(nC)) (matching the best cost-scaling bound known). We develop cost-scaling push-relabel implementations that take advantage of the assignment problem's special structure, and compare our codes against the best codes from the literature. The results show that the push-relabel method is very promising for practical use."
Author: Augustine Kwanashie
Publisher:
ISBN:
Category : Algorithms
Languages : en
Pages : 175
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Author: Josep Diaz
Publisher: Springer Science & Business Media
ISBN: 3642130720
Category : Computers
Languages : en
Pages : 394
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Book Description
This book constitutes the refereed proceedings of the 7th International Conference on Algorithms and Computation, CIAC 2010, held in Rome, Italy, in May 2010. The 30 revised full papers presented together with 3 invited papers were carefully reviewed and selected from 114 submissions. Among the topics addressed are graph algorithms I, computational complexity, graph coloring, tree algorithms and tree decompositions, computational geometry, game theory, graph algorithms II, and string algorithms.
Author: Aranyak Mehta
Publisher:
ISBN: 9781601987181
Category : Computers
Languages : en
Pages : 120
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Book Description
Matching is a classic problem with a rich history and a significant impact on both the theory of algorithms and in practice. Recently, there has been a surge of interest in the online version of matching and its generalizations. This is due to the important new application domain of Internet advertising. The theory of online matching and allocation has played a critical role in designing algorithms for ad allocation. Online Matching and Ad Allocation surveys the key problems, models, and algorithms from online matchings, as well as their implication in the practice of ad allocation. It provides a classification of the problems in this area, an introduction into the techniques used, a glimpse into the practical impact, and ponders some of the open questions that will be of interest in the future. Matching continues to find core applications in diverse domains, and the advent of massive online and streaming data emphasizes the future applicability of the algorithms and techniques surveyed here. Online Matching and Ad Allocation is an ideal primer for anyone interested in matching, and particularly in the online version of the problem, in bipartite graphs.
Author: Eugene Lawler
Publisher: Courier Corporation
ISBN: 048614366X
Category : Mathematics
Languages : en
Pages : 404
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Book Description
Perceptive text examines shortest paths, network flows, bipartite and nonbipartite matching, matroids and the greedy algorithm, matroid intersections, and the matroid parity problems. Suitable for courses in combinatorial computing and concrete computational complexity.
Author: Dan Gusfield
Publisher:
ISBN: 9780262515528
Category : Combinatorial analysis
Languages : en
Pages : 0
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Book Description
This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms.The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities.The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms.Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer.
