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This report summarizes work done as part of the Wavelet Based Image Analysis PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module provides mathematical background on the maxcut problem and develops an exact branch and bound algorithm for the maximum cut of unweighted graphs that is designed for improved performance on sparse graphs.

Introduction

Finding the maximum cut of a graph is a difficult to compute problem in combinatorial optimization with several applications in the world of engineering and physics. This research develops and evaluates an exact branch and bound algorithm for the maximum cut of unweighted graphs that is designed for improved performance on sparse graphs.

The module provides a general overview of the problem along with necessary mathematical background in "The Maxcut Problem" and a brief note on various approaches to the problem in "Several Algorithms" . "A New Algorithm" describes a new algorithm for finding maximum cuts. Results of empirical performance evaluation appear in "Empirical Testing" , which "Conclusion" further discusses.

The maxcut problem

Before discussing the maxcut problem, it is necessary to provide some background information regarding relevant concepts in graph theory, the most fundamental of which is the graph itself. A graph G ( V , E ) is an ordered pair comprised of a set of vertices V and a set of edges E that connect pairs of distinct vertices in V . Two examples are shown in [link] . Graphs may be either weighted, in which a real value is assigned to each edge, or unweighted, in which all edges have equal value. Although the former is more broadly applicable, further discussion will focus almost exclusively on the latter.

Two example graphs appear above.

Unsurprisingly, a subgraph G 1 ( V 1 , E 1 ) of graph G ( V , E ) is a graph with vertex set V 1 V and edge set E 1 E . Of particular usefulness will be the subgraph of G ( V , E ) induced by a given set of edges E 1 E , known as edge induced subgraph, which consists of that given set of edges E 1 along with all vertices V 1 = v | ( u , v ) E 1 , u V that occur as an endpoint of at least one edge in E 1 . An example of an edge induced subgraph is shown in [link] .

The subgraph induced by the red colored edges is shown on the right.

One class of graphs that will be especially important to discussion of the maxcut problem is bipartite graphs. A graph G ( V , E ) is bipartite, like the example in [link] , if there are sets V 1 , V 2 V such that V 1 V 2 = V , V 1 V 2 = , and ( u , v ) E only if u V 1 , v V 2 or v V 1 , u V 2 . Additionally, a graph is bipartite if and only if it has no subgraph that is a cycle of odd length.

In the above bipartite graph, the vertices are colored red or blue to highlight the vertex partitions.

A cut of a graph can be informally understood and visualized as a closed curve crossing some realization of the graph where each edge can be crossed at most once, as seen in [link] . Notice that the curve partitions the graph vertices into two disjoint subsets located to each side of the curve.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Lily
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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