upenn 502-05 - CIS 502 - Spring 2005 Final and WPE Exam:...

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Unformatted text preview: CIS 502 - Spring 2005 Final and WPE Exam: 5/03/05 1. (20 points) (a) Prove that if there is a fully polynomial-time approximation scheme for vertex cover, then P = NP. Answer: In a n vertex graph, the size of a minimum vertex cover is between 0 and n . If there is a fully polynomial approximation scheme for vertex cover, we can choose = 1 / ( n + 1) and get an algorithm that produces a vertex cover of size at most OPT (1+ 1 n +1 ). Since OPT is the largest integer less than or equal to this size, the algorithm must in fact produce an optimal vertex cover. Furthermore its running time is polynomial in n and 1 / = ( n + 1), which is polynomial in the input size. Thus, we would have a polynomial time algorithm for an NP-hard problem which shows that P = NP. (b) Let G be an arbitrary graph. Let M 1 and M 2 be any two maximal matchings in G . Prove that | M 1 | ≥ | M 2 | 2 . (Recall that the size of a matching M is denoted | M | and is equal to the number of edges in M .) Answer: There are many good proofs for this problem. One that I read in several of your papers is as follows: Suppose for contradiction that M 1 and M 2 are maximal matchings with | M 1 | < | M 2 | / 2. For each edge ( u,v ) in M 2 at least one of the endpoints must be matched by M 1 . Otherwise, we could have added ( u,v ) to M 1 contradicting the assumption that it is maximal. Each edge in M 1 covers two vertices and hence the | M 1 | edges can only hit 2 | M 1 | < | M 2 | of the pairs of vertices matched by M 2 . Another elegant proof from one of you: Let k be the size of the optimal vertex cover. From class we know that | M 1 | ≤ k ≤ 2 | M 1 || M 2 | ≤ k ≤ 2 | M 2 | Combining the first inequality from the first line and the second from the second we get | M 1 | ≤ 2 | M 2 | and symmetrically | M 2 | ≤ 2 | M 1 . 1 2. (20 points) The problem Half Clique is defined below. Problem: Half Clique Instance: Graph G = ( V,E ) with an even number of nodes....
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upenn 502-05 - CIS 502 - Spring 2005 Final and WPE Exam:...

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