spring 2015 mathematics in management science conflict scheduling vertex coloring chromatic number...
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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 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
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 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.
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.
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.
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
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