1. Special Graphs: Modeling and Algorithms

2. Flow graph modeling
Often, recognizing a task as network flow is biggest challenge.
Example:
Some number of users brings a set of applications (label ‘A’ to ‘Z’) to be run. There are 10 computers that can run applications (1 per day). Applications can only run on certain computers
Create nodes for each application, each computer
Create single source and single sink
Create edge from source to each application node
Set weight as the number of that application submitted by users
Create edge from application node to computer nodes that can run it
Weight can be 1, since only 1 program can run per computer
Create edge from computer nodes to the sink
Weight is 1, since each can run one program.

3. Another example (UVA problem 11380) – Titanic sinking
Grid of values: water (impassible) – ice with a person on it (will disappear once they leave), ice a person can move to (but will sink once they leave), iceberg (can hold people but not permanently), large wood (can hold people for safety, but only limited number)
Create graph connecting adjacent nodes, high edge capacity
Set node capacity for ice to 1
Create a source to all nodes with starting people (high edge weight)
Create a sink from all wood nodes with weight equal to max. wood capacity.

4. Bipartite graph and Bipartite matching
Divide into two groups, A and B
All edges are from something in group A to something in group B
Bipartite matching
Want to uniquely match one item from group A with one item in group B
Called Maximum Cardinality Bipartite Matching (MCBM)
Notice:
Bipartite matching must have bipartite graph where |A| = |B|
Given a bipartite graph, can do analysis to find a MCBM

5. Maximum Cardinality Bipartite Matching
Assign a source with an edge to every vertex in set 1
Assign a sink with an edge from every vertex in set 2
Run Max Flow
If it can find flow of amount equal to |A| (and therefore |B|), we have a match
If sets are unequal size, will find the best possible match (all of one should match to something from other, if possible)

6. Max Independent Set and Min Vertex Cover
Independent Set: no two vertices in set have edge between them
Vertex Cover: every vertex is adjacent to some vertex in the set
Number of vertices in min vertex cover is number in MCBM
Number of vertices in max independent set is total number minus number in MCBM
Note that finding these is a little different (but can use the MCBM matching for this)

7. MCBM with Augmenting Paths
Start at a free vertex (on “left”)
Find a free edge to a vertex on “right”. Assume it is already matched.
Follow the matched edge back to the “left”
Repeat until reaching a free vertex on the right
Then, reverse the order of all edges visited (from “free” <-> “matched”
Easy to implement (relatively) and fast enough on most cases.

8. Edge-disjoint paths
Paths in graphs – how many paths do not share an edge
Convert each edge to a unit-weight edge
If the graph is undirected, form two edges for each (one per direction)
Max flow in the graph is the number of edge-disjoint paths

9. Circulation Each node has either supply or demand, edges have capacity
Create a new source with edges to the nodes with supply
Edge weights are the amount of supply (negated)
Create a new sink with edges from the nodes with demand
Edge weights are the amount of demand
Then, just run max flow!
Several problems can be modeled as circulation problems

10. Other Graph Types DAG (Directed Acyclic Graph): Tree Eulerian Graph
Graph with an Eulerian path or tour: visits each edge exactly once
Several algorithms can be simplified if the graph is of one of these types
See book for examples