Chromatic Number of a Graph (Graph Colouring)
- 1. DAA PRESENTATION BY YASH BRID 2019130008 ABHISHEK CHOPRA 2019130009 ADWAIT HEGDE 2019130019
- 2. DECISION PROBLEM CHROMATIC NUMBER CHROMATIC NUMBER IS THE MINIMUM NUMBER OF COLORS REQUIRED TO COLOR ANY GRAPH SUCH THAT NO TWO ADJACENT VERTICES OF IT ARE ASSIGNED THE SAME COLOR. PROBLEM STATEMENT
- 3. What are NP Problems? NP is set of decision problems whose solutions are hard to find but easy to verify and can be solved by a Non-deterministic Turing Machine in Polynomial time. A Non-deterministic Turing Machine is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations i.e. the set of rules of a Non-deterministic Turing Machine may prescribe more than one action to be performed for any given situation. A problem X is NP-Complete if there is an NP problem Y, such that Y is reducible to X in polynomial time. NP-Complete problems can be solved by a Non-deterministic Turing Machine in polynomial time. What is a Non-deterministic Turing Machine? What is a NP-Complete problem? PROBLEM DEFINITION
- 4. What is a NP-Hard Problem? Satisfiability with at most 3 literals per clause. Reduction to NP-Hard problem is possible It is a reduction from 3-SAT. The complexity class of decision problems that are intrinsically harder than those that can be solved by a nondeterministic Turing machine in polynomial time. When a decision version of a combinatorial optimization problem is proved to belong to the class of NP- complete problems, then the optimization version is NP-hard. Therefore, it is a NP-Hard Problem. Why is CNDP a NP-Hard Problem?
- 5. Graph Colouring: Graph Coloring is the process of assigning colors to the vertices of a graph such that no two adjacent vertices of it are assigned the same color. THEORY Chromatic number: Chromatic Number is the minimum number of colors required to color any graph such that no two adjacent vertices of it are assigned the same color. Example:
- 6. ALGORITHM Create a recursive function that takes current index, number of vertices and output color array. If the current index is equal to number of vertices. Check if the output color configuration is safe. If the conditions are met, print the configuration and break. Assign color to a vertex (1 to m). For every assigned color recursively call the function with next index and number of vertices. If any recursive function returns true break the loop and return true. There are quite a few ways in which we can solve the Chromatic Number Decision Problem (CNDP), backtracking and naive are two of them. Naive: 1. 2. 3. 4. 5. 6.
- 7. Create a recursive function that takes the graph, current index, number of vertices and output color array. If the current index is equal to number of vertices. Print the color configuration in output array. Assign color to a vertex (1 to m). For every assigned color, check if the configuration is safe, recursively call the function with next index and number of vertices. If any recursive function returns true break the loop and return true If no recusive function returns true then return false. Backtracking: 1. 2. 3. 4. 5. 6.
- 13. SOME EXAMPLES
- 14. APPLICATION Map Colouring Sudoku Register Allocation Time Table Scheduling Mobile Radio Frequency
- 15. MAP COLOURING Four colors are sufficient to color any map We place a vertex in each region 4-coloring thereom
- 16. Fill in the blank cells so that each row, column and box has the characters 1 to 9 exactly once SUDOKU
- 17. How do we schedule the exam so that no two exams with a common student are scheduled at same time? How many minimum time slots are needed to schedule all exams? TIMETABLE MAKING
- 18. CNDP is an np-hard problem NP-hard are still under research, so until these are found one will have to use the algorithm with exponential time complexity (naive/backtracking) CONCLUSION
- 20. QUESTIONS