Ford-Fulkerson Method (Chapter 10)

• Notice that you need several steps (3?).
• At the end, to find the cut, you need to calculate the final \(G_f\).
• See Jamboard.

Reductions to Maxflow Problems

• Can solve using Ford-Fulkerson in \(O(E|f^*|)\) time.
• Can solve using really fancy algorithms in \(O(VE)\) time.
• So if you can translate a given problem to a maxflow problem, you can solve it in \(O(X + VE)\) time, where the translation process takes \(O(X)\) time.
  • This is called reducing the given problem to a maxflow problem.
• Example: Hostile Dogs reduces to maxflow.
  • Translation is \(O(E)\). (Put weights on edges.)
  • \(|f^*|\) can be taken to be 2. (Stop when you reach 2.)
  • Hostile dogs time: \(O(E|f^*|) = O(E)\). (Can just use FF.)

Bipartite Matching

• A bipartite graph is an undirected graph whose vertex set consists of two partitions, \(L\) and \(R\), such that every edge goes from \(L\) to \(R\).
  • i.e., a bipartite graph is 2-colorable.
• A bipartite matching is a selection of edges in a bipartite graph that doesn’t duplicate vertices.
  • The edges match a vertex in \(L\) with a vertex in \(R\).

• Example. Given a bipartite graph where \(L\) represents workers \(R\) represents jobs, and an edge from a worker to a job means the worker can do that job:
  • Assign each worker a job.
  • Do so maximally (assign as many jobs as possible)
• Bipartite matching reduces to a maximum flow problem.

Group Exercise

The flow network on the Jamboard purports to solve a bipartite matching problem by finding a maximum flow. Show that this really is a maximum flow by finding a minimum cut.

Tuple Selection

• Extension of bipartite matching.
• Vertex set partitioned: \(X_1, X_2, X_3, \ldots\).
• Each \(X_i\) is a different finite resource.
• Edges represent constraints among these resources.

Example: Scheduling Finals

You have to assign rooms, times, and proctors for final exams in 9 different classes. There are 5 rooms, 4 exam times, and 7 proctors.

• Each class has one final exam.
• Each class has a list of rooms that are big enough.
• Each proctor has a list of times they are available.
• No proctor should have to give more than 5 exams.

See Figure 11.5.

Scheduling Finals Time

• Let \(N\) be the total number of classes, rooms, times, proctors combined.
• \(O(N)\) vertices, \(O(N^2)\) edges.
• Translating takes \(O(E) = O(N^2)\) time.
• Computing max flow takes \(O(VE) = O(N^3)\) time, using Orlin’s algorithm.
  • Could also just use Ford-Fulkerson, because \(|f^*|\) is bounded by the number of classes.
• Time: \(O(N^2 + N^3) = O(N^3)\).

Baseball Elimination

Can a team still finish with the most wins?

• You know how many games each team has left to play.
• You know how many wins each team has.
• You know which team has to play which other team.

See Figure 11.7.

Baseball elimination time?

Suppose there are \(n\) teams. The flow model has a start vertex \(s\), a list of pairing vertices, a list of team vertices, and a terminal vertex \(t\).

1. How many possible pairings of teams are there? (Give an exact answer, and an asymptotic estimate.)

2. How many vertices are in the flow model? (asymptotically)

3. How many edges leave each pairing vertex? How many edges are in the flow model?

4. What is the running time, using Orlin’s algorithm?

5. Would using the Floyd-Fulkerson algorithm ever have a lower asymptotic running time?

Disjoint Path Covers

• You are given a dag.
  • e.g., vertices are tasks with some prerequisite constraints.
• What is the fewest number of paths in the dag that cover all the vertices?
  • e.g., the tasks in each path can be performed by the same person.

Reduces to bipartite matching (which reduces to maxflow).

See Figure 11.6.