Definition of Karp's polynomial-time transformation: Problem X polynomial transforms to problem Y if given any input x to X, we can construct an input y such that x is 'yes' instance of X iff y is yes instance of Y. 从定义可以看出， problem X与 problem Y之间的对应关系是 any input x to an input y。我在一开始的学习过程中，没有体会到这个定义的含义，所以对arbitrary problem Y有要求，其实如果any input x to any input y，两者是可以等价的，拆分成any input x to an input y, and any input y to an input x.
可以从vertex cover, independent set, hitting set,set cover之间的规约关系来理解这层定义。
(1)From vertex cover to independent set, let S be any vertex cover, consider two nodes u and v in V - S, note that no edge between vertex (u,v), thus no nodes in V-S connected to each other. V-S is independent set.
(2)From vertex cover to hitting set. Given a vertex cover 'G(V,E),k'. We can construct subsets S_1,S_2,S_3...S_n and hitting set H from set U (U is a set contains all the elements). In this case, a vertex cover is arbitrary, so we use the clause:"Given a vertex ", but hitting set is a specific construction. In order to meet the demand for the corresponding relationship between arbitrary vertex cover and specific construction, let each subset S_1,S_2,S_3...S_n has just two elements {u,v}, and (u,v) represents an edge in G(V,E)(in other words, S_i represent an edge in Graph). But one point should be proposed that the proof is necessary and sufficent condition to meet phrase 'if and only if' in definition.
(3)set cover，we can see attached figure. The specific construcition of problem Y uses the rules:"(1)each element appears only twice in total because element represents nodes in graph, and each node just has two endpoints.or we can make edge e_i can exist in different pair of nodes.(2)Just one edge between each pair of nodes, which means number of common elements in different subsets doesn't exceed 1. " So from the rules we can know this is a one-way reduction, just form vertex cover to set cover. In other words, two problem is not in equivalence.
我在理解问题的时候找到了一个网站可以作为这个问题理解的参考，网址：https://cs.stackexchange.com/questions/76176/reduction-from-set-cover-to-vertex-cover
