In the world of graph theory, bipartite graphs play a significant role. A bipartite graph is an undirected graph where its nodes can be divided into two sets, and edges only exist between those two sets. In this article, we will explore the concept of perfect matching in bipartite graphs and how it can be implemented using SQL Server.

Understanding Bipartite Graphs

Before diving into perfect matching, let’s first understand what bipartite graphs are. A bipartite graph consists of two sets of nodes, often referred to as the left and right nodes. The edges in the graph only connect nodes from different sets. For example, consider the following bipartite graph:

What is Perfect Matching?

In a bipartite graph, a perfect matching is a collection of edges that share no nodes and for which every node is part of some edge in that collection. The goal is to find a perfect matching in a given bipartite graph or determine if one cannot exist.

The Algorithm

The script presented in this article implements a popular algorithm for finding perfect matchings in bipartite graphs. The algorithm works by first randomly selecting as many disjoint edges as possible until no more can be found. It then modifies the collection of edges repeatedly until it covers all nodes or shows that a perfect match is not possible.

The algorithm follows these steps:

1. Apply the greedy algorithm to get a partial matching.
2. If the partial match fails to match at least half of all nodes, return with an error.
3. If the match is perfect, return with the result.
4. Call the initialization of S.
5. While 1 = 1:
• Call the augmentation of M repeatedly until augmentation fails.
• If augmentation succeeds:
• If the match is now perfect, return with the result.
• Call the initialization of S.
• Call the expansion of S since augmentation of M just failed.
• If unable to expand S, no perfect match is possible, so return with an error.

Implementing the Algorithm in SQL Server

The algorithm can be implemented in SQL Server using stored procedures. The script calls four stored procedures: spGreedy, spInitializeS, spExpandM, and spExpandS. These stored procedures handle different aspects of the algorithm, such as applying the greedy algorithm, initializing the sets, expanding the matching, and expanding the sets.

Here is an example of how the stored procedure spExpandM works:

ALTER PROCEDURE [dbo].[spExpandM]
AS
BEGIN
    -- Code for spExpandM goes here
END

The other stored procedures follow a similar structure, each handling a specific part of the algorithm.

Testing the Algorithm

To test the algorithm, you can use the provided test script. The script randomly generates a bipartite graph with a specified number of left nodes and a probability for edge creation. It then applies the algorithm to find a perfect matching.

Here is an example of how to use the test script:

-- Declarations
DECLARE @Result int

-- No counting
SET NOCOUNT ON

-- Build random bipartite graph with selected number of left nodes
-- and probability for edge creation.
EXEC @Result = spBuildBipartite 10, 0.25
IF @Result <> 1 RETURN

-- Apply the algorithm to find a perfect matching
EXEC @Result = spFindPerfectMatching
IF @Result <> 1 RETURN

The test script generates a random bipartite graph and then applies the algorithm to find a perfect matching. You can adjust the number of left nodes and the probability for edge creation to explore different scenarios.

Conclusion

In this article, we explored the concept of perfect matching in bipartite graphs and how it can be implemented using SQL Server. The provided algorithm and stored procedures offer a way to find perfect matchings in bipartite graphs or determine if one cannot exist. By understanding and implementing these concepts, you can solve various problems related to bipartite graphs using SQL Server.

References:

1. Graph Theory
2. Discrete Mathematics by Laszlo Lovasz
3. Hall’s marriage theorem