Geometry of Cuts and Metrics

Deza, Michel; Laurent, Monique

doi:10.1007/978-3-642-04295-9

Cited by 759 publications

(981 citation statements)

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“…This cone is called the cut cone for reasons which we will soon see. A whole book is dedicated to the basic properties of this cone and its relatives [DL97].…”

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

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,509

1,355

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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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“…This cone is called the cut cone for reasons which we will soon see. A whole book is dedicated to the basic properties of this cone and its relatives [DL97].…”

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

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,509

1,355

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“…Many classes of strong valid inequalities have been discovered for CUT ; see again [9,17]. Here, we are interested in the gap inequalities of Laurent and Poljak [19], which take the following form:…”

Section: Literature Reviewmentioning

confidence: 99%

“…The maxcut problem is a fundamental and well-known combinatorial optimization problem, proven to be strongly -hard in [12]. It has a surprisingly large number of important practical applications, and has received a great deal of attention (see, e.g., the book [9] and the survey [17]). …”

Section: Introductionmentioning

confidence: 99%

“…Such inequalities can then be exploited algorithmically within a branch-and-cut framework (see, e.g., [6]). A wide array of such inequalities have been discovered for the max-cut problem (see again [9]). In particular, Laurent and Poljak [19] introduced an intriguing class of strong inequalities, known as gap inequalities, which includes several other known classes as special cases.…”

Section: Introductionmentioning

confidence: 99%

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Complexity results for the gap inequalities for the max-cut problem

Galli

Kaparis

Letchford

2012

Operations Research Letters

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In 1996, Laurent and Poljak introduced an extremely general class of cutting planes for the max-cut problem, called gap inequalities. We prove several results about them, including the following: (i) there must exist non-dominated gap inequalities with gap larger than 1, unless = co-; (ii) there must exist non-dominated gap inequalities with exponentially large coefficients, unless = co-; (iii) the separation problem for gap inequalities can be solved in finite time (specifically, doubly exponential time).

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“…There are two good reasons for this approach. First, it is possible, in principle, to formulate a property-preserving mapping to a reduced dimension space [9,14,16]. Second, we can take advantage of existing sequence similarity algorithms for metric comparison (see [24] and included references).…”

Section: Performing Organization Name(s) and Address(es)mentioning

confidence: 99%

Characteristic Shape Sequences for Measures on Images

Pinge¹,

Abramson²,

Asaki³

et al. 2006

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Researchers in many fields often need to quantify the similarity between images using metrics that measure qualities of interest in a robust quantitative manner. We present here the concept of image dimension reduction through characteristic shape sequences. We formulate the problem as a nonlinear optimization program and demonstrate the solution on a test problem of extracting maximal area ellipses from twodimensional image data. To solve the problem numerically, we augment the class of mesh adaptive direct search (MADS) algorithms with a filter, so as to allow infeasible starting points and to achieve better local solutions. Results here show that the MADS filter algorithm is successful in the test problem of finding good characteristic ellipse solutions from simple but noisy images.

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Geometry of Cuts and Metrics

Cited by 759 publications

References 0 publications

Expander graphs and their applications

Expander graphs and their applications

Complexity results for the gap inequalities for the max-cut problem

Characteristic Shape Sequences for Measures on Images

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