1 In this problem, we will define the Set Cover problem and show that it is NP-complete. (a) (5 points) Suppose I gave you some set U = {1,2,3,4,5} of elements and some subset of U’s power set S = {{1,2,3).(24).(3,4),(4,5). Is ther choose at most 1 set in S that covers all of U? How about if I let you pick 2 sets? 3? No 12 3 (b) (5 points) Show that SETCOVER-(U,S,ISCP(U) and there is some subset T C S of size k that covers all of U) E NP. You may pick whichever definition of NP you prefer. (c) (10 points) Show that VERTEXCOVER Sp SETCoVER. That is, given some graph G and integer k, we need to construct some U and S and k such that (G, kje VERTEXCOVER仁キ〈U, s,k’) e SETCoVER, and this construction is limited to a polynomial amount of time in KG, Show transcribed image text 1 In this problem, we will define the Set Cover problem and show that it is NP-complete. (a) (5 points) Suppose I gave you some set U = {1,2,3,4,5} of elements and some subset of U’s power set S = {{1,2,3).(24).(3,4),(4,5). Is ther choose at most 1 set in S that covers all of U? How about if I let you pick 2 sets? 3? No 12 3 (b) (5 points) Show that SETCOVER-(U,S,ISCP(U) and there is some subset T C S of size k that covers all of U) E NP. You may pick whichever definition of NP you prefer. (c) (10 points) Show that VERTEXCOVER Sp SETCoVER. That is, given some graph G and integer k, we need to construct some U and S and k such that (G, kje VERTEXCOVER仁キ〈U, s,k’) e SETCoVER, and this construction is limited to a polynomial amount of time in KG,

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