Exam 2 Date #
Exam 2 will be Next Thursday, April 9th
What Good is Min Flow / Max Cut? #
Edge-Disjoint Paths #
Problem: Find the number of distinct paths (no shared edges) from s to t.
Solution:
- Make this a weighted graph with every edge having weight 1.
- Now max flow = # of distinct paths.
Vertex-Disjoint Paths #
Problem: How about no shared intermediate vertices?
Solution:
- Replace each vertex with two with a weight 1 edge between them.
- Now max flow = # of distinct paths.
Bipartite Matching #
Example:
- We’ve got a set of doctors and a set of hospitals.
- Doctors have hospitals they will work for, and hospitals have doctors they will hire.
- We want to maximize number of hires.
Solution:
- Construct a graph:
- Doctors are vertices.
- Hospitals are vertices.
- There’s an edge from each doctor to each hospital where both doctor and hospital accept the other. Each wt = 1.
- Add vertex s with edge to each doctor, wt = 1.
- Add vertex t with edge from each hospital, wt = |V|.
- Calculate max flow.
Each hospital only wants one doctor? h -> t weight is 1.
Tuple Selection #
Finding assignments of several “resources”.
Let’s give concrete example:
Exam Selection #
Flow network:
- s -> classes -> rooms -> times -> proctors -> t
Specifically:
- An edge s 0 c i with capacity 1 for each class i . (“Each class can hold at most one final exam.”)
- An edge c i r j with capacity 1 for each class i and room j where the class will fit in the room capacity.
- An edge r j t k with capacity 1 for each room j and time slot k . (“At most one exam can be held in room j at time k.”)
- An edge t k p with capacity 1 for time slot k and proctor such that
A[, k] = T. (“A proctor can oversee at most one exam at any time, and only during times that they are available.”) - An edge p -> t 0 with capacity 5 for each proctor (“Each proctor can oversee at most 5 exams.”)
Example Problems #
- Question 11.4
- Question 11.11
