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Chapter 8: Q15E (page 280)
Show that the following problem is NP-complete.
MAXIMUM COMMON SUBGRAPHInput: Two graphs and ; a budget .Output: Two set of nodes and whose deletion leaves at least nodes in each graph, and makes the two graphs identical.
The MAXIMUM COMMON SUBGRAPH is NP-complete.
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Sequencing by hybridization. One experimental procedure for identifying a new DNA sequence repeatedly probes it to determine which (substrings of length ) it contains. Based on these, the full sequence must then be reconstructed.Let鈥檚 now formulate this as a combinatorial problem. For any string x (the DNA sequence), let denote the multiset of all of its localid="1658905204605" . In particular, localid="1658904556515" contains exactly elements.The reconstruction problem is now easy to state: given a multiset of strings, find a string x such that is exactly this multiset.
Show that the reconstruction problem reduces to RUDRATA PATH. (Hint: Construct a directed graph with one node for each localid="1658904858295" , and with an edge from a to b if the last characters of match the first localid="1658905395287" characters of b.)
But in fact, there is much better news. Show that the same problem also reduces to EULER PATH. (Hint: This time, use one directed edge for each .)
Consider a special case of 3SAT in which all clauses have exactly three literals, and each variable appears exactly three times. Show that this problem can be solved in polynomial time. (Hint: create a bipartite graph with clauses on the left, variables on the right, and edges whenever a variable appears in a clause. Use Exercise 7.30 to show that this graph has a matching.)
STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer K , find a satisfying assignment in which at most K variables are true, if such an assignment exists. Prove that is -complete.
Show that if then the RSA cryptosystem (Section 1.4.2) can be broken in polynomial time.
Search versus decision. Suppose you have a procedure which runs in polynomial time and tells you whether or not a graph has a Rudrata path. Show that you can use it to develop a polynomial-time algorithm for RUDRATA PATH (which returns the actual path, if it exists).
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