Vertex cover reduces to set cover

Posted 2022-01-09 mb61caa1c74a413

目录

• 1.1 Vertex cover
  • 1.1.1 VERTEX-COVER ≤ P SET-COVER
  • 1.1.2 证明充分性
  • 1.1.3 证明必要性

1.1 Vertex cover

例子：
U = 1, 2, 3, 4, 5, 6, 7
Sa = 3, 7 Sb = 2, 4
Sc = 3, 4, 5, 6 Sd = 5
Se = 1 Sf= 1, 2, 6, 7
k = 2（Sc和Sf可以覆盖完原集合U）

1.1.1 VERTEX-COVER ≤ P SET-COVER

Pf. Given a VERTEX-COVER instance G = (V, E) and k, we construct a
SET-COVER instance (U, S, k) that has a set cover of size k iff G has a
vertex cover of size k.

Construction：

• Universe U = E.
• Include one subset for each node v ∈ V : Sv = e ∈ E : e incident to v .
  
  

1.1.2 证明充分性

Pf. ⇒ Let X ⊆ V be a vertex cover of size k in G.
Then Y = Sv : v ∈ X is a set cover of size k.
“no” instances of VERTEX-COVER are solved correctly！

一个实列：

如果我们选择f与c结点那么恰好可以对应到一个子集合覆盖。
但是如果我们顶点覆盖的点选择f和d,那么对应的子集合覆盖为Sf和sd显然这两个集合不能覆盖原集合U。
结论：无充分性！

1.1.3 证明必要性

Pf. ⇐ Let Y ⊆ S be a set cover of size k in (U, S, k).

• Then X = v : Sv ∈ Y is a vertex cover of size k in G.

“yes” instances of VERTEX-COVER are solved correctly！