1. Problem Solving 4

2. Vertex Cover in Bipartite Graph
Given a bipartite graph, find a vertex cover with minimum cardinality.

3. König's theorem

4. König's theorem Max Flow = Min Cut

5. Algorithm

6. Maximum-Weight Matching
Given a (bipartite) graph with edge weight, find a matching with maximum total weight.

7. Disjunct Matrix

8. Problem

9. Lemma
Consider a collection C of pools of size at most 2. Let G be the graph with all items as vertices and all pools of size 2 as edges. Then
C gives a d-disjunct matrix if and only if every item not in a singleton pool has degree at least d+1 in G.

10. Proof

12. Solution of the Problem

13. Remark
This problem can be solved in polynomial-time.

14. Augmenting Path

15. Optimality Condition
The Proof is similar to that on maximum matching.