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**Bipartite Matching, Extremal Problems, Matrix Tree Theorem.**

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**Today’s Plan Proof of Hall’s theorem Algorithms for bipartite matching**

Extremal problems Matrix tree theorem

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**Perfect Matching Does a perfect matching always exist? Of course not.**

If there are more vertices on one side, then of course it is impossible. N(S) S Let N(S) be the neighbours of vertices in S. If |N(S)| < |S|, then it is impossible to have a perfect matching.

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**A Necessary and Sufficient Condition**

Is it the only situation when a bipartite graph does not have a perfect matching? Hall’s Theorem: A bipartite graph G=(V,W;E) has a perfect matching if and only if |N(S)| >= |S| for every subset S of V and W. This is a deep theorem. It tells you exactly when a bipartite graph does not have a perfect matching. (Now you can convince the king.)

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**Proof of Hall’s Theorem (easy direction)**

Hall’s Theorem: A bipartite graph G=(V,W;E) has a perfect matching if and only if |N(S)| >= |S| for every subset S of V. One direction is easy, if there is a perfect matching, then |N(S)| >= |S| for every subset S of V. S N(S) Just consider the neighbours of S in the perfect matching.

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**Proof of Hall’s Theorem (difficult direction)**

Hall’s Theorem: A bipartite graph G=(V,W;E) has a perfect matching if and only if |N(S)| >= |S| for every subset S of V. Another direction is more interesting, we need to show whenever |N(S)| >= |S|, then there is a perfect matching! How to prove such kind of statement? Proof by strong induction on the number of edges.

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**Proof of Hall’s Theorem**

Hall’s Theorem: If |N(S)| >= |S| for every subset S of V, then there is a perfect matching. Case 1: Every subset S has |N(S)| > |S|. (Easy case) Just delete an edge. By deleting an edge, |N(S)| can decrease by at most 1. Since |N(S)| > |S| before, we still have |N(S)| >= |S| after deleting an edge. Since the graph is smaller (one fewer edge), by induction, there is a perfect matching in this smaller graph, hence there is a perfect matching in the original graph.

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**Proof of Hall’s Theorem**

Hall’s Theorem: If |N(S)| >= |S| for every subset S of V, then there is a perfect matching. Case 2: Suppose there is a subset S with |N(S)| = |S|. Divide the graph into two smaller graphs G1 and G2 (so we can apply induction) Find a perfect matching in G1 by induction. G1 Then we are done. G2 S N(S) Find a perfect matching in G2 by induction.

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**Proof of Hall’s Theorem**

Why there is a perfect matching in G2? Hall’s Theorem: If |N(S)| >= |S| for every subset S of V, then there is a perfect matching. To apply Hall’s, we want to show for any subset T of S, |N(T) G2| >= |T|. T |N(T) G2| G2 S N(S)

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**Find a perfect matching in G2 by induction.**

Proof of Hall’s Theorem Why there is a perfect matching in G2? For any subset T S, N(T) G2 = N(T). By assumption, |N(T) G2| = |N(T)| >= |T|. Therefore, by induction, there is a perfect matching in G2. T |N(T) G2| G2 S N(S) Find a perfect matching in G2 by induction.

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**Proof of Hall’s Theorem**

Why there is a perfect matching in G1? For any subset T, we want to show |N(T) G1| >= |T| to apply induction. Consider T, by assumption, |N(T)| >= |T| Can we conclude that |N(T) G1| >= |T|? No, because N(T) may intersect N(S)! Now what? N(T) G1 T S N(S)

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**Proof of Hall’s Theorem**

Why there is a perfect matching in G1? For any subset T, we want to show |N(T) G1| >= |T| to apply induction. Consider S T, by assumption, |N(S T)| >= |S T| (the green areas). Since |S|=|N(S)|, |N(S T) – N(S)| >= |S T - S| (the red areas). So |N(T) G1| = |N(S T) – N(S)| > = |S T – S| = |T|, as required. N(T) G1 T |S|=|N(S)| S N(S)

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**Proof of Hall’s Theorem**

Hall’s Theorem: If |N(S)| >= |S| for every subset S of V, then there is a perfect matching. Case 2: Suppose there is a subset S with |N(S)| = |S|. Divide the graph into two smaller graphs G1 and G2 (so we can apply induction) Find a perfect matching in G1 by induction. G1 Now we are done. G2 S N(S) Find a perfect matching in G2 by induction.

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**Today’s Plan Proof of Hall’s theorem Algorithms for bipartite matching**

Extremal problems Matrix tree theorem

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First Try Greedy method? (add an edge with both endpoints unmatched)

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Key Questions How to tell if a graph does not have a (perfect) matching? How to determine the size of a maximum matching? How to find a maximum matching efficiently?

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**Existence of Perfect Matching**

Hall’s Theorem [1935]: A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A. N(S) S

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**Bound for Maximum Matching**

What is a good upper bound on the size of a maximum matching? König [1931]: In a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover. König [1931]: In a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover. Min-max theorem NP and co-NP Implies Hall’s theorem.

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Algorithmic Idea? Any idea to find a larger matching?

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Augmenting Path Given a matching M, an M-alternating path is a path that alternates between edges in M and edges not in M. An M-alternating path whose endpoints are unmatched by M is an M-augmenting path.

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**Optimality Condition What if there is no more M-augmenting path?**

If there is no M-augmenting path, then M is maximum! Prove the contrapositive: A bigger matching an M-augmenting path Consider Every vertex in has degree at most 2 A component in is an even cycle or a path Since , an M-augmenting path!

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**Algorithm Key: M is maximum no M-augmenting path**

How to find efficiently? How to find efficiently?

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**Finding M-augmenting paths**

Orient the edges (edges in M go up, others go down) An M-augmenting path a directed path between two unmatched vertices

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**Complexity At most n iterations**

An augmenting path in time by a DFS or a BFS Total running time

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**Minimum Vertex Cover Hall’s Theorem [1935]:**

A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A. König [1931]: In a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover. Idea: consider why the algorithm got stuck…

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**Today’s Plan Proof of Hall’s theorem Algorithms for bipartite matching**

Extremal problems Matrix tree theorem

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