1. Movie theatre service on brightness and volume range leading to maximum clique graph By, Usha Kavirayani

2. OUTLINE Problem statement Intersection Graphs of Boxes Problem Solution Graph Construction Maximum Clique History NP-Complete Graph Techniques of dealing NP-Complete graphs References

3. DO YOU WATCH MOVIES???

4. 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.

5. 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

6. 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.

7. 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.

8. 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 160-8020-40 Person 255-7030-45 Person 350-5535-65 Person 440-5555-80 Person 535-4570-90 Person 660-7075-95 Person 770-8575-90 Person 860-8080-100

9. Volume Brightness

10. Graph Construction Depending on the given brightness and volume ranges we shall draw a graph Here the maximum clique is {6,7,8}

11. MAXIMUM CLIQUE Clique Clique : A clique is a set of pairwise adjacent vertices Maximum clique: The maximum clique problem is to find the clique number, , of a graph, i.e., the size of the largest clique in the graph ω(H) = size of maximum clique of H Maximal Clique Maximal Clique: A clique that cannot be enlarged by adding any more vertices

12. Maximum Clique of Size 5 EXAMPLE

13. 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.

14. 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.

15. 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

16. Techniques for Dealing with NP-complete Problems The main techniques to solve NP-complete problems are- Backtracking Branch and Bound

17. 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.

18. 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.

19. 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.

20. 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

21. REFERENCES http://www.eprisner.de/Papers/AJourneyThroughIntersectionGraphCounty.pdf 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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