Geometric Algorithms and Combinatorial Optimization

Grötschel, Martin; Lovász, László; Schrijver, Alexander

doi:10.1007/978-3-642-78240-4

Cited by 1,151 publications

(1,381 citation statements)

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“…By the same reasoning, such a problem can be turned into a problem of convex optimization, in which we are trying to optimize some objective function on the cut cone. There is a general theory of discrete optimization as laid out in [GLS93], in which we try to optimize some linear function over a convex domain Ω. In order to use this machinery we must be able to solve efficiently two basic questions for Ω: (i) Membership -To determine, given a point x, whether it belongs to Ω; and (ii) Separation -Same as above, but if x ∈ Ω, find a hyperplane that separates x from Ω.…”

Section: S|(n − |S|) ≥ H(g) Nmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,502

1,386

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: S|(n − |S|) ≥ H(g) Nmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,502

1,386

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“…The value of that relaxation is called the theta number, and is denoted by ϑ (G) for a graph G. The theta number arises from several different formulations, see Grötschel, Lovász, and Schrijver [1988]. Van Hoeve [2006] uses the formulation that has been shown to be computationally most efficient among those alternatives [Gruber and Rendl, 2003].…”

Section: Problem Description and Model Formulationsmentioning

confidence: 99%

“…Van Hoeve [2006] uses the formulation that has been shown to be computationally most efficient among those alternatives [Gruber and Rendl, 2003]. Let us introduce that particular formulation (called ϑ 3 by Grötschel, Lovász, and Schrijver [1988]). Let x ∈ {0, 1} n be the vector of binary variables representing a stable set, where n = |V |.…”

Section: Problem Description and Model Formulationsmentioning

confidence: 99%

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Semidefinite Programming and Constraint Programming

Hoeve

2011

International Series in Operations Research &Amp; Management Science

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“…Grötschel et al [9][10][11] generalize the notion of an odd set and define a triple family as follows.…”

Section: Triple Families: a Generalization Of Even/oddmentioning

confidence: 99%

Fast Algorithms for Even/Odd Minimum Cuts and Generalizations

Benczúr

Fülöp²

2000

Algorithms - ESA 2000

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Abstract. We give algorithms for the directed minimum odd or even cut problem and certain generalizations. Our algorithms improve on the previous best ones of Goemans and Ramakrishnan by a factor of O(n) (here n is the size of the ground vertex set). Our improvements apply among others to the minimum directed Todd or T-even cut and to the directed minimum Steiner cut problems. The (slightly more general) result of Goemans and Ramakrishnan shows that a collection of minimal minimizers of a submodular function (i.e. minimum cuts) contains the odd minimizers. In contrast our algorithm selects an n-times smaller class of not necessarily minimal minimizers and out of these sets we construct the odd minimizer. If M (n, m) denotes the time of a u-v minimum cut computation in a directed graph with n vertices and m edges, then we may find a directed minimum -odd or T-odd cut withThe key of our construction is a so-called parity uncrossing step that, given an arbitrary set system with odd intersection, finds an odd set with value not more than the maximum of the initial system.

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Geometric Algorithms and Combinatorial Optimization

Cited by 1,151 publications

References 0 publications

Expander graphs and their applications

Expander graphs and their applications

Semidefinite Programming and Constraint Programming

Fast Algorithms for Even/Odd Minimum Cuts and Generalizations

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