算法设计期末作业03-8.10

题目

Proving NP-completeness by generalization. For each of the problems below, prove that it is NP-complete by showing that it is a generalization of some NP-complete problem we have seen in this chapter.

　　(a) SUBGRAPH ISOMORPHISM: Given as input two undirected graphs G and H, determine whether G is a subgraph of H (that is, whether by deleting certain vertices and edges of H we obtain a graph that is, up to renaming of vertices, identical to G), and if so, return the corresponding mapping of V (G) into V (H).

　　(b) LONGEST PATH: Given a graph G and an integer g, find in G a simple path of length g.

　　(c) MAX SAT: Given a CNF formula and an integer g, find a truth assignment that satisfies at least g clauses.

　　(d) DENSE SUBGRAPH: Given a graph and two integers a and b, find a set of a vertices of G such that there are at least b edges between them.

　　(e) SPARSE SUBGRAPH: Given a graph and two integers a and b, find a set of a vertices of G such that there are atmost b edges between them.

(f) SET COVER. (This problem generalizes two known NP-complete problems.)

　　(g) RELIABLE NETWORK: We are given two n×n matrices,a distance matrix dij and a connectivity requirement matrix rij, as well as a budget b; we must find a graph G = ({1,2,…,n},E) such that (1) the total cost of all edges is b or less and (2)between any two distinct vertices i and j there are rij vertex-disjoint paths. (Hint: Suppose that all dij’s are 1 or 2, b = n,and all rij’s are 2. Which well known NP-complete problem is this ?)

问题解答

（a）
这道题我们只需要把CLIQUE问题规约到子图同构问题即可。不失一般性，我们给定G和一个完全图K，下面只需要证明（K，G）的实例的解是正确的当且仅当G有一个g个顶点的团,即去证明K是不是G的子图的问题。若K是G的子图，即给出（K,G）的一个肯定解，G有一个g个顶点的团。故必要性得证。若G 有一个g个顶点的团，g个顶点的完全图也是G的子图，故综上所述，得证。

（b）
其实就相当于找一条顶点数为g+1的哈密顿路，二者是对应的关系。

（c）
g表示句子的总数，显然MAX SAT和这个SAT是对应的关系

（d）
这道题其实是把CLIQUE规约成稠密子图的问题。令b=a(a-1)/2,由（a）显然（K，a,b）是稠密子图的一个实例解，当且仅当G中有一个g个顶点构成的团，且这个g个顶点是一个完全图，边数为a(a-1)/2。

（e）
这道题其实是把独立集规约成稀疏子图的问题。对于独立集问题（G，k）,可以令a=k, b=0，于是任意两个顶点之间没有边，规约成功。

（f）
这道题其实是从点覆盖规约成集合覆盖。不失一般性，对于点覆盖问题(V, E)，令集合覆盖问题中的K（最多可选的集合数），为点覆盖问题中最多可选边数，令所有集合元素的并集是点覆盖问题中的边集，令S（v）为和顶点v相邻的边的集合，故找点覆盖可以规约到找S（v）集合的集合覆盖。

（g）
这道题其实是TSP问题规约到可靠网络问题。即它是TSP问题的推广，可知它也是NPC的。