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Transcript of Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha...
![Page 1: Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha Kavirayani.](https://reader038.fdocuments.us/reader038/viewer/2022110320/56649cc25503460f9498aa3f/html5/thumbnails/1.jpg)
Movie theatre service on brightness and
volume range leading to maximum clique
graph
By,Usha Kavirayani
![Page 2: Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha Kavirayani.](https://reader038.fdocuments.us/reader038/viewer/2022110320/56649cc25503460f9498aa3f/html5/thumbnails/2.jpg)
OUTLINE
• Problem statement
• Intersection Graphs of Boxes
• Problem Solution
• Graph Construction
• Maximum Clique
• History
• NP-Complete Graph
• Techniques of dealing NP-Complete graphs
• References
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DO YOU WATCH
MOVIES???
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PROBLEM STATEMENT
• A new service at a movie theatre is to ask every viewer the range of brightness and the range of sound volume he or she would accept.
• Every person in the theatre has their own set of requirements on brightness and volume ranges.
• Some prefer low volume, while some like to enjoy watching a movie with high volume.
![Page 5: Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha Kavirayani.](https://reader038.fdocuments.us/reader038/viewer/2022110320/56649cc25503460f9498aa3f/html5/thumbnails/5.jpg)
PROBLEM STATEMENT (CONT)
• In similar way every person wants to watch a movie with different brightness range.
• According to these votes the projectionist chooses that adjustment that satisfies most people. How does he find it?
• This problem can be solved using maximum clique graph.
• This problem is NP-complete for general graphs
![Page 6: Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha Kavirayani.](https://reader038.fdocuments.us/reader038/viewer/2022110320/56649cc25503460f9498aa3f/html5/thumbnails/6.jpg)
Intersection Graphs of Boxes
• As for many geometric interaction models, the first step, finding sufficiently many star graphs is easy for interaction graphs of boxes in R
• We only have to find the cliques.
• The problem is the layout step, where we have to place the cliques on the plane in a certain way.
• After the placement has been done for every vertex x of G, the cliques containing x generate a smallest axis parallel rectangle, which we denote by Sx.
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Cont…
• The obey-
For every vertex x of G, all cliques in Sx must contain x
If Sx and Sy intersect, then they have some clique in common.
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PROBLEM SOLUTION
• Let us consider 8 people watching a movie
• Each person has different volume range
• Each person has different brightness range
Person number
Brightness range
Volume range
Person 1 60-80 20-40
Person 2 55-70 30-45
Person 3 50-55 35-65
Person 4 40-55 55-80
Person 5 35-45 70-90
Person 6 60-70 75-95
Person 7 70-85 75-90
Person 8 60-80 80-100
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Volume
Brightness
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Graph Construction
• Depending on the given brightness and volume ranges we shall draw a graph
Here the maximum clique is {6,7,8}
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MAXIMUM CLIQUE
• Clique : A clique is a set of pairwise adjacent vertices
• Maximum clique: The maximum clique problem is to find the clique number, w, of a graph, i.e., the size of the largest clique in the graph
• ω(H) = size of maximum clique of H
• Maximal Clique: A clique that cannot be enlarged by adding any more vertices
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Maximum Clique of Size 5
EXAMPLE
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HISTORY
• The ‘Clique’ terminology comes from Luce and perry (1949).
• First Algorithm for solving the Clique problem is that of Harary and Ross (1957).
• Tarjan and Trojanowski (1977), an early work on the worst-case complexity of the Maximum Clique problem
• In the 1990s, a breakthrough series of papers beginning with Feige (1991) and reported at the time in major newspapers, showed that it is not even possible to approximate the problem accurately and efficiently.
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NP-COMPLETE GRAPH
• P. Problems that can be solved in polynomial time. ("P" stands for polynomial.) These problems have formed the main material of this course
• NP. This stands for "nondeterministic polynomial time" where nondeterministic is just a fancy way of talking about guessing a solution. A problem is in NP if you can quickly (in polynomial time) test whether a solution is correct (without worrying about how hard it might be to find the solution). Problems in NP are still relatively easy: if only we could guess the right solution, we could then quickly test it.
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NP-COMPLETE GRAPH(CONT)
• The “hardest” problems in NP are called NP-complete problems (NPC)
• Why “hardest”? A problem X is NP-complete if and only if-
1.X is in NP and
2.Any problem Y in NP can be converted to an instance of X in polynomial time, such that solving X also provides a solution for Y
In other words: Can use algorithm for X as a subroutine to solve Y
• Thus, if you find a poly time algorithm for just one NPC problem, all problems in NP can be solved in poly time
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Techniques for Dealing with NP-complete Problems
• The main techniques to solve NP-complete problems are-• Backtracking
• Branch and Bound
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BACKTRACKING
• Backtracking- Explore possibilities; backtrack when doesn’t work.
• This backtracking algorithm is a method for finding all the subsets in an undirected graph G.
• Given a graph G with ‘V’ vertices and ‘E’ edges, G = (V, E)
• Let us take an integer variable k.
• This algorithm is used in scientific and engineering applications.
• This algorithm is a Depth First Search algorithm.
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• The algorithm for finding k-clique in an undirected graph is a NP-complete problem.
• List out all the possibilities in the sub graph and check for each and every edge.
• Check for a sub graph in which every node is connected to every other node.
• Check for all possible Cliques in the graphs.
• Check the size of clique whether it is equal to k or not.
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BRANCH AND BOUND
• Branch n bound: Variation for case where finding minimum (or maximum) of objective function
• Where backtracking uses a depth-first search with pruning, the branch and bound algorithm uses a breadth-first search.
• Starting by considering the root node and applying a lower-bounding and upper-bounding procedure to it.
• If the bounds match, then an optimal solution has been found and the algorithm is finished
• If they do not match, then algorithm runs on the child nodes.
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• Upper bound: e.g. a feasible solution
• Lower bound:
e.g. a solution to an “easier” problem
• Node elimination:
when lower bound >= upper bound
Example – Traveling salesman problem
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REFERENCES
• http://www.eprisner.de/Papers/AJourneyThroughIntersectionGraphCounty.pdf
• NP-complete - Wikipedia, the free encyclopedia
• http://ise.tamu.edu/people/faculty/butenko/papers/EORMS.pdf
• http://homepages.warwick.ac.uk/~masgax/Graph-Theory-notes.pdf
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ANY QUERIES???
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THANKYOU