算法NP－complete Ex8.3

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Ex8.3

题目：
In the HITTING SET problem, we are given a family of sets{$S_{1},S_{2},...,S_{n}$} and a budget b, and we wish to find a set H of size $\le$ b which intersects every $S_{i}$, if such an H exists. In other words, we want $H\bigcap S_{i} \neq \emptyset$for all i.
Show that HITTING SET is NP-complete.

Ans:
很容易将最小顶点覆盖归约到 HITTING SET。假设要求图 G 的最小顶点覆盖,可以 建立一个 HITTING SET 实例,其中{$S_{1},S_{2},...,S_{n}$}即是图G的各条边,比如
{$v_{1}, v_{2}$},{$v_{3}, v_{4}$}, 。通过二分式的询问,可以找到一个与Si都相 交的最小集合 H ,这正好就是图 G 的最小顶点覆盖。