Spring 2015 Mathematics 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 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 overlap FMHPEISC French X XXX X Mathematics X XX History XXX Philosophy X X English XX X Italian XXX X X Spanish X X Chemistry X XX

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 Colorings Sometimes 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.

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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 is L = # edges = # of vtxs. Every circuit can be colored using 2 or 3 colors. The chromatic number of a circuit is CN = 2 if even length, CN = 3 if odd length.

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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 K N 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 length CN = 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 overlap FMHPEISC French X XXX X Mathematics X XX History XXX Philosophy X X English XX X Italian XXX X X Spanish X X Chemistry X XX

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