Algorithms_Part15 - S. Dasgupta, C.H. Papadimitriou, and...

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Unformatted text preview: S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 281 In this case, ingredients 2 and 3 go together pretty well whereas 1 and 5 clash badly. Notice that this matrix is necessarily symmetric; and that the diagonal entries are always . . Any set of ingredients incurs a penalty which is the sum of all discord values between pairs of ingredients . For instance, the set of ingredients { 1 , 3 , 5 } incurs a penalty of . 2 + 1 . 0 + 0 . 5 = 1 . 7 . We want this penalty to be small. E XPERIMENTAL CUISINE Input: n , the number of ingredients to choose from; D , the n × n “discord” matrix; some number p ≥ Output: The maximum number of ingredients we can choose with penalty ≤ p . Show that if EXPERIMENTAL CUISINE is solvable in polynomial time, then so is 3 SAT . 8.17. Show that for any problem Π in NP , there is an algorithm which solves Π in time O (2 p ( n ) ) , where n is the size of the input instance and p ( n ) is a polynomial (which may depend on Π ). 8.18. Show that if P = NP then the RSA cryptosystem (Section 1.4.2) can be broken in polynomial time. 8.19. A kite is a graph on an even number of vertices, say 2 n , in which n of the vertices form a clique and the remaining n vertices are connected in a “tail” that consists of a path joined to one of the vertices of the clique. Given a graph and a goal g , the KITE problem asks for a subgraph which is a kite and which contains 2 g nodes. Prove that KITE is NP-complete. 8.20. In an undirected graph G = ( V, E ) , we say D ⊆ V is a dominating set if every v ∈ V is either in D or adjacent to at least one member of D . In the DOMINATING SET problem, the input is a graph and a budget b , and the aim is to find a dominating set in the graph of size at most b , if one exists. Prove that this problem is NP-complete. 8.21. Sequencing by hybridization. One experimental procedure for identifying a new DNA sequence repeatedly probes it to determine which k-mers (substrings of length k ) it contains. Based on these, the full sequence must then be reconstructed. Let’s now formulate this as a combinatorial problem. For any string x (the DNA sequence), let Γ( x ) denote the multiset of all of its k-mers. In particular, Γ( x ) contains exactly | x | - k + 1 elements. The reconstruction problem is now easy to state: given a multiset of k-length strings, find a string x such that Γ( x ) is exactly this multiset. (a) Show that the reconstruction problem reduces to R UDRATA PATH . ( Hint: Construct a di- rected graph with one node for each k-mer, and with an edge from a to b if the last k- 1 characters of a match the first k- 1 characters of b .) (b) But in fact, there is much better news. Show that the same problem also reduces to E ULER PATH . ( Hint: This time, use one directed edge for each k-mer.) 8.22. In task scheduling, it is common to use a graph representation with a node for each task and a directed edge from task i to task j if i is a precondition for j . This directed graph depicts the....
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Algorithms_Part15 - S. Dasgupta, C.H. Papadimitriou, and...

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