Spring 2015Mathematics in

Management Science

Conflict Scheduling

Vertex Coloring

Chromatic Number

Conflict Resolution

Resolving Conflicts

Another type of scheduling problem: here want to avoid conflicts in scheduling. For example:

block exams—classes with overlapping students need exams the diff times

equipment—jobs needing same equip can’t be scheduled at the same time

Displaying Conflict Data

Have data describing (potential) conflicts (or lack of) btwn things.

Exs fish, animals in zoo, final exams

Can show via conflict table.• Have columns and rows.• An X means ‘conflict’ (or not).

Can show via conflict graph.

Conflict Table

Way to display conflicts.

X in column/row means ‘conflict’.

A, B & A, D & B, C all “conflicted”.

Note symmetry!

Example

Block exam conflicts for fall term

(excluding language exams).

Example Scheduling Exams

Have eight exams to schedule: French, Math, History, Philosophy, English, Italian, Spanish, Chemistry

Some students are taking two or more classes.

Only two air-conditioned rooms.

Conflict Table X if overlapF M H P E I S C

French   X   X X X   XMathematics X       X X    History           X X XPhilosophy X             XEnglish X X       X    Italian X X X   X   X  Spanish     X     X    Chemistry X   X X        

Conflict Graphs

Can represent conflicts with a graph:• vertices—things to schedule• vertices connected by an edge if two

have a conflict (i.e, can’t be scheduled at the same time).

Easy to read this off conflict table.

Conflict Graph Example

Pix’d table gives graph.

Useful to “clean up” graph.

Conflict Graph

Two vtxs connected by an edge need different schedule times.

Use (different) colors for times!

Try to color the vtxs so that any two vtxs connected by an edge have different colors.

This called a vertex coloring of graph.

Vertex Coloring Example

Vertex Colorings

Can always use a different color for each vtx (i.e. schedule everything for unique times), but this is not efficient!

What is fewest number of colors (times) can use to get a sched w/o conflicts?

Vertex ColoringsSometimes have limit on number of

items that can be scheduled at the same time.

This corresponds to limit on number of vertices with same color.

E.g., only 4 rooms available for exams, means can’t schedule more than 4 at the same time.

Vertex Coloring

Color all vertices of graph so that any two vertices joined by an edge have different colors.

The minimum number of colors needed is the chromatic number of the graph.

Example

Example •

Example

Example

Vertex Coloring

Minimum number of colors needed (to have a vertex coloring) is the chromatic number of the graph.

To see that a graph has chromatic number CN, must show:• there is vtx coloring with NC colors,• cannot color with less than NC colors.

Coloring Circuits

The length of a circuit isL = # edges = # of vtxs .

Every circuit can be colored using 2 or 3 colors.

The chromatic number of a circuit isCN = 2 if even length,CN = 3 if odd length.

Example•

• •

Example

Coloring Complete Graphs

A graph is complete if every pair of vtxs is joined by an edge.

Any vertex coloring of a complete graph with N vertices must use N different colors.

The chromatic number of KN is

CN = N .

Useful Fact

If a graph has a subgraph with chromatic number N, then the bigger graph will have chromatic number at least N. (Can’t use fewer colors!)

This useful when a bigger graph has a smaller complete graph built into it.

Example

Brooks’ Theorem

G a graph which is not complete nor a circuit (of odd length)

G’s chromatic number satisfies

CN ≤ maximum vertex valence.

Chromatic Number

Minimum number of colors need.The chromatic number of a cplt

graph is CN = # of vtxs .The chromatic number of a circuit

CN = 2 if even lengthCN = 3 if odd length

All other graphs have CN at most the maximum vertex valence .

Example Scheduling Exams

Have eight exams to schedule: French, Math, History, Philosophy, English, Italian, Spanish, Chemistry

Some students are taking two or more classes.

Only two air-conditioned rooms.

Conflict Table X if overlapF M H P E I S C

French   X   X X X   XMathematics X       X X    History           X X XPhilosophy X             XEnglish X X       X    Italian X X X   X   X  Spanish     X     X    Chemistry X   X X        

Conflict Graph

• classes correspond to vertices• edges join conflicted vertices• look for vertex coloring

any two vertices joined by an

edge have different colors• colors are different exam times

Conflict Graph