train depot problem using permutation graphs by venkatesh pasunuri

TRAIN DEPOT PROBLEM USING PERMUTATION GRAPHS

By

Venkatesh Pasunuri

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

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

A REAL-WORLD PROBLEMThe 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.

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.

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

B

C

A

B C

Directed Graph

Comparability Graph

Transitive

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.

BASIC PROPERTIES

*Transitive orientation is present for both.

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.*

BASIC PROPERTIESComplements of permutation graphs are permutation graphs.

Complement

Permutation Graphs

Permutation Graphs are self Complementary

CONSTRUCTING A GRAPH

Ordering ProblemPermutation Graph

Equivalent

•Assigning a train according to the order they appear can be written geometrically in a form of a permutation graph.

Minimum coloring of a general graph is NP-completeIt becomes linear on a permutation graph and can be, Solved in linear manner.

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.

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.

SPECIAL PROPERTY

Permutation graph

SOLUTION TO TRAIN DEPOT PROBLEM

Solution:

• 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]

Working view of problem

Assuming Every Line can accommodate two trains

TRAIN DEPOT PROBLEM CONT…


Final view at night



Trains leaving in the morning



Trains leaving in the morning

Finally, all the trains are out of depot.


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 .

•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.

CONCLUSION

REFERENCES• Martin Charles Golumbic, Algorithmic graph theory and

perfect graphs, Annals of Discrete Mathematics, Elsevier, vol. 57, 2004 (2nd edition).

• Wikipedia

• https://kluedo.ub.uni-kl.de/files/2263/marshall.pdf

THANK YOU !!!!!

QUERIES ?

train depot problem using permutation graphs by venkatesh pasunuri

Documents

introduction permutation

given graph

permutation symmetry

undirected graph g

realworld problem

scheduling problem

offline problem

online problem