NP 问题证明详解 set cover

在论文On Network Correlated Data Gathering附录二中的过程极其详细，于是我深入拜读以后，写下这篇总结！

首先了解一下set cover问题是什么问题？

Instance:
A collection C of subsets of a finite set P and an interger 0<K<|C|, with |C| the cardinality of C.
Question: Does C contain a subset of C' \in C with |C'|<K such that every element of P belongs to at least one of the subset C'?

The Set Cover problem is: Given a set of elements E = {e1, e2, . . . , en} and a set of m subsets of E, S = {S1, S2, . . . , Sn}, find a “least cost” collection C of sets from S such that C covers all
elements in E. That is, ∪Si∈CSi = E

然后了解一下我们的问题：Network data gathering tree cost problem

Instance: A undirected graph G=(V,E) with weights d_{i,j} assigned to the edges(i,j)\ in E, a positive integer M, and a particular node S \in V.
Question: Does the graph admit a spanning tree ST such that when assigning supplies R_i=R to the leaf nodes and R_i=r<R to the in-tree nodes in the spanning tree ST, the
total cost of ST given by(7) is ata most M?

For any instance of set cover problem, we build an instance of our decision problem. Fig. 1 illustrates the construction of the graph instance for our problem.



For each element C_i, we build a structure formed by 4 nodes x1, x2, x3, x4 as in the following figure:



关于graph中权重，权重d, a, 1都如图所示，all other edges are assumed of infinite weight and are not plotted.

我们的目标是找到一个spanning tree for this graph, for which the cost in (7) is at most M. we now show that if M=...for the positive interger K<|C|, then finding a spanning tree with the cost at most M is equivalent
to finding a set cover of cardinality K or less for the set P.