Graph Theory: using Bipartite matching and Edge coloring

1. Prepared By
Niaz Mohammad

2. Combinatorial problem
Class Scheduling
Linear Programming
Class Scheduling is a broad general
problem which has been one of the prime Optimization Problem
area of research. There has been a lot of
algorithms developed for this particular Genetic Algorithm
problem using different techniques as
shown on the right.
Constraint Propagation
This is a problem with so many variations
due to various constraints that can be tied
to it. And this is why it is so hard and is Simulated Annealing
considered as NP(Non-Polynomial)
complete i.e. it does not have any
Tabu Search
polynomial time bound.
However, approximate algorithm or
heuristic approach can be taken to attain Greedy Algorithm
solution in a feasible time bound.
Hill Climbing

3. Problem Statement
Design a simple class schedule
with no conflict between a set
of teachers and a set of
courses with the following set
of constraints:
i) No same courses can be
taken by any teacher or
batch for a given time
slot.
ii) Minimize the number of
time slots required and
prove that it is optimum.

4. My Class-scheduling Approach
I researched for this problem a lot and have found
numerous articles, documents that were published on
this topic and have seen so many variants of this Bipartite Matching
problem and also various approaches such as GA(genetic
algorithm), CSP (Constraint Satisfaction
Problem), Combinatorial approaches have been taken.
But for my purpose of creating or developing the
simplest form of this class-scheduling problem, I tried to Edge Coloring
develop my own algorithm, it works !
Well, for at least the specific example that Jolly madam
presented in our class.
I put the teachers and the courses into two independent
sets and then created the Bipartite matching according
to the matrix of number of courses taken by each
teacher.
Draw the timetable
So, in a nutshell, the idea is to find the maximum
Bipartite matching among teacher & courses and then
do an edge coloring to find the time slot so that no
two edges coming out from a vertex have the same
color.

5. Edge Coloring Model
• Take bipartite graph with Teacher/Batch 21st 22nd 23rd 24th
vertices for teachers and for RAJ 2 0 1 1
classes AR 1 1 0 0
• Look for a coloring of the SA 0 0 2 1
edges such that no vertex SIS 0 2 0 2
has two incident edges with KMH 1 1 1 0
the same color.
• What is the minimum
number of colors needed?
– Lower bound: maximum
degree. (Interpretation!)
– We can attain the lower bound
with help of matching!!

7. A Theorem
Let G be a bipartite graph with
maximum degree d. Then G has an
edge coloring with d colors.
 Step 1: Make G regular
by adding vertices and
edges.
 Step 2: Repeatedly find
a matching and remove
it.

8. Edge coloring a regular graph
Say G’ is regular of degree d.
For i = 1 to d do
 Find a perfect matching
M in G’.
 Give all edges in M color
i.
 Remove all edges in M
from G’. (Note that G’
stays regular!)

9. Final step
Take the edge coloring c of G’. Color
G in the same way: G is subgraph of
G’.
Time: carrying out d times a perfect
matching algorithm in a regular
graph:
 O(nd3) if we use
Schrijver’s algorithm.
 Can be done faster by
other algorithms.

10. My Algorithm
1. Find the maximum number of
degree(d) from the given bipartite Time Complexity:
graph.
O(nd3)
2. Add vertex and edges so that it Where n is the number of
becomes a d-regular bipartite vertices(in teacher’s set(t)) and d is
graph. the maximum number of degree.
3. Split if there are cycles in the
given Bipartite graph until there
are no cycle.
4. For each color c from 1 to d
 Iteratively color different batch
for set t ( the edges between
them) with color c greedily.
5. Represent it as final matrix which
is the solution.