Problem Solving 4

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

König's theorem

König's theorem Max Flow = Min Cut

Algorithm

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

Disjunct Matrix

Problem

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.

Proof

Solution of the Problem

Remark This problem can be solved in polynomial-time.

Augmenting Path

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