Matching Problem Examples

Graph Theory · Axiom Academy

EXAMPLE Matching Problem Examples Master maximum matchings, Hall's condition, perfect matchings, and augmenting paths Problem Setup: Job Assignment Problem Excellent work! You've mastered the fundamentals of matching problems in bipartite graphs. Here's what we learned: Maximum Matching: A matching M is maximum if no other matching has more edges. We use augmenting paths to find maximum matchings - if no augmenting path exists, the matching is maximum. Augmenting Paths: An alternating path from an unmatched vertex in X to an unmatched vertex in Y, where edges alternate between "not in M" and "in M". Flipping edges along this path increases the matching size by 1. Hall's Marriage Theorem: A bipartite graph G(X,Y) has a matching that saturates X if and only if |N(S)| ≥ |S| for every subset S ⊆ X. This is called Hall's condition. Perfect Matching: When |X| = |Y| and the maximum matching has size |X|, we have a perfect matching - every vertex is matched. Perfect matchings exist if and only if Hall's condition holds for both X and Y. Berge's Theorem: A matching M is maximum if and only if there is no M-augmenting path. This gives us a polynomial-time algorithm for finding maximum matchings. König's Theorem: In a bipartite graph, the size of a maximum matching equals the size of a minimum vertex cover. This beautiful duality connects matchings to covering problems. These matching algorithms are used in job assignments, network flows, DNA sequencing, and many optimization problems!

This is the written version of the interactive lesson above. See the full Graph Theory course.