11.1.3. Perfect Match and Network Flows

So, here is a problem which seems to our nothing to do with flow. So, we have some course allocation to be done in a school, we have a collection of teachers and we have a collection of courses and each teacher as indicated the preference of which courses he or she is willing to teach.

In this part, we are introduced to a practical problem of allocating courses to teachers in a school. Each teacher has specific courses they are willing to teach, forming a set of preferences. The goal is to match each teacher to a course satisfying their preferences.

Imagine a group of students selecting clubs to join based on their interests. Each student has preferences for which clubs they want to join, and the task is to assign them to clubs in such a way that their interests are respected, similar to how we will match teachers to courses.

So, this is what is called as a matching problem, in particular it is called bipartite matching. So, what is a bipartite graph? Bipartite graph is which one in which the vertices come in two groups which we called v0 and v1 and all the edges go from v0 to v1.

In a bipartite graph, nodes are divided into two distinct sets, and edges only connect nodes from one set to nodes in the other set. In our scenario, teachers represent one set (v0), while the available courses represent the second set (v1). This structure helps in visualizing relationships and constraints between the two sets.

Think of a dating app where users can swipe right for potential matches. Users form one set (v0), and the matches form another set (v1). The app creates connections only between users and potential matches, ensuring that relationships only exist between these two groups, similar to how teachers and courses are matched.

If we have an equal number of nodes on each side in a bipartite graph, then we can expect or hope that everything on the left is matched to something on the right. But each edge would match out two people.

A perfect match occurs when every teacher is assigned exactly one course, and each course is taught by exactly one teacher. This is attainable only if the number of teachers equals the number of courses, ensuring all preferences can be satisfied without overlap.

Consider a sports league draft where each team has the same number of player slots. If each player can only be picked by one team, a perfect match ensures every team is filled with enthusiastic players, and every player is on a team they wanted to join.

So, here is how we can answer this question using network flows. We add a spurious source node feeding into the teachers and we add a spurious target node, a sink node coming out of the courses.

To apply network flows to the matching problem, we introduce two extra nodes: a source that connects to teachers and a sink that connects from courses. Each connection (edge) has a capacity of one, indicating that one teacher can only be assigned one course, and vice versa.

Think of a water supply system where water flows from a reservoir to various faucets (the courses). Each faucet can only handle one stream of water at a time. By setting up baths for teachers and courses, we can visualize and manage the flow of assignments much like regulating water to prevent overflow and ensure that each faucet is served correctly.

So, the general example of what we called a reduction. So, we want to solve a given problem, problem A, but we do not know how to solve it, but we do know how to solve another problem B.

In algorithm design, a reduction is a technique used to relate two problems. By transforming the matching problem into a flow problem, we can utilize existing algorithms designed for network flow to find solutions. This is efficient because flow algorithms are well-studied and optimized.

Imagine needing to write an essay (Problem A) but finding that it's easier to create an outline (Problem B) first. By reducing the task of writing an essay to creating an outline, you can leverage tools and techniques for outlining to make the final essay easier to complete.

If I have an efficient solution for B and if this conversion process is efficient, then the efficiency of the algorithm for B compose with the efficiency of the pre-processing and the post-processing step will give us an efficient algorithm for A.

The efficiency of solving the original problem (A) depends on two factors: the efficiency of the solution for the transformed problem (B), and the efficiency of converting A to B and interpreting the solution from B back into A. This highlights the importance of both reduction and solution techniques in algorithm design.

Think of hiring a skilled contractor (efficient solution for B) to renovate a home (problem A). If the contractor not only does the work effectively but also follows a streamlined process to gather requirements and check on progress, the entire renovation process becomes quicker and more efficient.

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