算法作业 NP-complete problems 8.22

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本文探讨了反馈弧集问题及其与顶点覆盖问题之间的联系。反馈弧集问题要求在一个有向图中找到最小数量的边移除，使得图变得无环。文章通过构造一种特殊的有向图实例，证明了这两个问题间的等价性，并展示了如何从一个问题的解决方案推导出另一个问题的解。

8.22

In task scheduling,it is common to use a graph representation with a node for each task and a directed edge from task i to task j if i is a precondition for j.This directed graph depicts the precedence constraints in the scheduling problem.Clearly,a schedule is possible if and only if the graph is acyclic;if it isn't,we'd like to identify the smallest number of constraints that must be dropped so as to make it acyclic.

Given a directed graph G=(V,E),a subset E'≤E is called a feedback arc set if the removal of edges E' renders G acyclic.

FEEDBACK ARC SET(FAS):Given a directed graph G=(V,E) and a budget b,find a feedback arc set of <=b edges,if one exists.

(a)Show that FAS is in NP

FAScan be shown to be NP-complete by a reduction from VERTEX COVER.Given an instance (G,b) of VERTEX COVER,where G is an undiected graph and we want a vertex cover of size <=b,we construct a instance (G',b) of FAS as follows.If G=(V,E) has n vertices v1,...,vn,then make G'=(V',E') a directed graph with 2n vertices w1,w1',...,wn.wn',and n+2|E|(directed) edges:

1.(wi,wi') for all i=1,2,...,n.

2.(wi',wj) and (wj',wi) for every (vi,vj)∈E.

(b)Show that if G containsa vertex cover of size b,then G' contains a feedback arc set of size b.

(c)Show that if G' contains a feedback arc set of size b,then G contains a vertex cover of size (at most) b.(Hint:Given a feedback arc set of size b in G',you may need to first modify it slightly to obtain another one which is of a more convenient form,but is of the same size or smaller.Then,argue that G must contain a vertex cover of the same size as the modified feedback arc set.)

(a) 显然FAS是可在多项式时间内验证的，因此属于NP。
(b) 设G的一个大小为b的顶点覆盖为C，对于任意顶点vi∈C ，设其在 ' G中相对应的顶点为wi和wi ',则将边(wi,wi'),添加到 E'。对C中的每个顶点都这样处
理后，所得到的边集E'即是G'的一个大小为b的feedback arc set。因为对于顶点wi和wi',当去掉边(wi,wi')后，所有与wi相连的边都不可能位于任何一个
环中，因为wi不存在出边，同样，所有与wi'相连的边也不可能位于任何一个环中，因为wi'不存在入边。
(c) 对于G中的任意一条边(vi,vj),设其在G'中相对应的顶点为wi,wi',wj,wj',相对应的边为(wi,wi'),(wj,wj'),(wi',wj),(wj',wi). 若E'是G'的一个大小为b的feedback arc set，显然，在这四条边中至少有一条边e属于E'，否则就会形成环，而边e必然有个端点属于{wi,wj}.若wi是e的端点，则将vi加入到C，否则将vj加入到C。容易看出，在经过上述处理后，C即是G的一个大小不超过b的顶点覆盖。