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Unformatted text preview: CS3230 Tutorial 10 1. Consider the following problem: Input: Given a weighted graph G , two vertices u and v in G , and a value d . Question: Is there a path from u to v of weight at most d ? Is the above problem in NP? Could it be NP-complete? Ans: Yes, it is in P (and thus in NP), as we can solve the problem using Dijkstra’s algorithm in polynomial time. Yes, it could be NP-complete, if P=NP. If P 6 = NP, then it will not be NP-complete. 2. In class we saw that it is open at present whether P = NP or not. It is also open whether NP = EXP or not. Is it possible that both P = NP and NP = EXP are true? Ans: No. As that would imply P = EXP , which is known not to be true. 3. It can be shown that discrete knapsack problem is NP-complete. Thus, if discrete knap- sack problem can be solved in polynomial time, then all problems in NP can be solved in polynomial time. Professor S claimed that he could solve the discrete knapsack problem in time proportional to C * n (see the dynamic programming algorithm done in class), where C is the capacity of the knapsack and n is the number of objects in the problem. Thus the discrete knapsack problem is in P....
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- Fall '10