1. Train DEPOT PROBLEM USING PERMUTATION GRAPHSBy
Venkatesh Pasunuri

2. Agenda
A Real world problem Definition
Constructing a graph from a R-W problem
Why is it hard on general graphs
Special property
Solution to Train Depot Problem
Applications of permutation graphs
Conclusion
References

3. A Real-world problem: Train Depot Problem
Problem Definition
Arrange the trains on the minimum number of depot tracks
Put the trains on correct tracks to minimize shunting operations
Trains arrive at different times, stay overnight and leave the area in the morning
Train coming first not necessarily leaves first

4. A REAL-world PROBLEM The scheduling problem has two variance:
The offline problem:
This is the problem we are going to discuss here.
Here the depot knows the schedule of the trains in advance and calculations can be done prior to the trains arrival.
The online problem:
Here, the schedule of the trains is not known prior to their arrival.
A few trains are already in the depot and the new trains arrive without any prior information.
This variant of train problem is solved using greedy approach.
The trains are sent to the lines using the most suitable track at that time.

5. Introduction
PERMUTATION GRAPH In mathematics, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition.

6. Basic PROPERTIES
 Permutation graphs are also subclass of Comparability graphs
Comparability Graphs: Any undirected graph G that connects pairs of elements that are comparable to each other in a partial order.
These graphs are Transitive in orientation. A A
Transitive B C B C
Directed Graph
Comparability Graph

7. Basic PROPERTIES
Theorem: A graph G is a permutation graph if and only if both G and its complement G’ are both comparability graphs.
How to do it?
Find a transitive orientation of G and one of G’.
Lets assume G ≈G[], then G is a comparability graph since G[] has a transitive orientation.

8. Basic PROPERTIES
Now, according to the theorem and this property we can say that a graph is a permutation graph if, by applying the transitive orientation algorithm to it and its complements returns true.*
*Transitive orientation is present for both.

9. Permutation Graphs are self Complementary
Basic PROPERTIES
Complements of permutation graphs are permutation graphs.
Complement
Permutation Graphs are self Complementary
Permutation Graphs

10. CONSTRUCTING A GRAPH
Assigning a train according to the order they appear can be written geometrically in a form of a permutation graph.
Equivalent
Ordering Problem
Permutation Graph
Minimum coloring of a general graph is NP-complete
It becomes linear on a permutation graph and can be, Solved in linear manner.

11. Why is it hard on general graphs
->In case of using general graphs we cannot say whether they may reach the depot or not but in case of permutation graphs we can say that it will reach the depot at some point irrespective of time, because permutations can have only two output’s i.e; fail or pass. ->Interval graphs cannot be used for train depot problem because using the interval graphs there should be a connection (or) an interaction between two trains instead of that we are using permutation graphs in which we have only two possible outcome’s i.e: either pass or fail.

12. SPECIAL PROPERTY MINIMAL COLORING PROPERTY:
In graph theory , graph coloring is a special case of graph labelling ; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color.

13. SPECIAL Property
Permutation graph

14. Solution to Train Depot Problem
Lets assume we have N trains
Incoming trains-permutation [π1, π2, π3,…, πn] (here each train is represented as πi, I being an integer)
Outgoing Train Sequence S=[1,2,3,…,N]
[π1, π2,… πn]
Depot
S=[1,2,3,…N]

15. Train Depot Problem Cont…
Working view of problem
Assuming Every Line can accommodate two trains

16. Train Depot Problem Cont…
Working view of problem

17. Train Depot Problem Cont…
Working view of problem

18. Train Depot Problem Cont…
Working view of problem

19. Train Depot Problem Cont…
Final view at night
Working view of problem

20. Train Depot Problem Cont…
Trains leaving in the morning
Working view of problem

21. Train Depot Problem Cont…
Trains leaving in the morning
Working view of problem

22. Train Depot Problem Cont…
Trains leaving in the morning
Working view of problem

23. Train Depot Problem Cont…
Trains leaving in the morning
Working view of problem
Finally, all the trains are out of depot.

24. Applications of permutation graphs
Permutation graphs have numerous applications in various fields of study
As they are a subclass of perfect graphs, many problems can be solved efficiently which are NP-complete on arbitrary graphs.
A few examples include:
Clique cover
Treewidth via dynamic programming on scan lines
Weighted Independent domination problems
Independent set problems
Domination clique problems
There are many practical implementations of permutation graphs .

25. Conclusion
Permutation graphs are both comparability and co comparability graphs.
These are subclasses of perfect graphs with a lot of variants.
They can also be viewed as circle graphs in some cases.
Permutation graphs are useful to solve a lot of graph problems.
These graphs have numerous real life applications too.
Many optimization problems become polynomial on permutation graphs.

26. References
Martin Charles Golumbic, Algorithmic graph theory and perfect graphs, Annals of Discrete Mathematics, Elsevier, vol. 57, 2004 (2nd edition).
Wikipedia

27. THANK YOU !!!!!
QUERIES ?