算法概论课本题目Chapter 8 Exercises 8.10

8.10. 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 edgeso f 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,findi n 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 ?)

解：

a) 令图G 为一个环，环上的顶点数等于图H 的顶点数。那么若G 是H 的同构子 图，则说明H 存在 Rudrata 回路。

于是知 Rudrata 回路事实上是子图同构问题的 一个特例。
b) 如果令 g = |V| −1，即得到一条 Rudrata 路径。
c) 令 g 为子句的总数，即成 SAT。
d) 令 b = a*(a-1)/2，此时这a个顶点两两相连，于是即成大团问题。
e) 令 b = 0，即成大独立集问题。

f) 显然是小顶点覆盖的一个推广。

g) Hint 中所描述的特例即是一个 TSP。