Question 1 |

The problem 3-SAT and 2-SAT are

both in P | |

both NP complete | |

NP -complete and in P respectively | |

undecidable and NP complete respectively |

Question 1 Explanation:

Question 2 |

Language L_{1} is polynomial time reducible to language L_{2}. Language L_{3} is polynomial time reducible to L_{2}, which in turn is polynomial time reducible to language L_{4}. Which of the following is/are true?

I. if L_{4} \in P, then L_{2} \in P

II. if L_{1} \in P or L_{3} \in P, then L_{2} \in P

III. L_{1} \in P, if and only if L_{3} \in P

IV. if L_{4} \in P, then L_{1} \in P and L_{3} \in P

I. if L_{4} \in P, then L_{2} \in P

II. if L_{1} \in P or L_{3} \in P, then L_{2} \in P

III. L_{1} \in P, if and only if L_{3} \in P

IV. if L_{4} \in P, then L_{1} \in P and L_{3} \in P

II only | |

III only | |

I and IV only | |

I only |

Question 2 Explanation:

Question 3 |

Consider two decision problems Q1,Q2 such that Q1 reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to Q2. Then which one of the following is consistent with the above statement?

Q1 is in NP, Q2 is NP hard. | |

Q2 is in NP, Q1 is NP hard | |

Both Q1 and Q2 are in NP . | |

Both Q1 and Q2 are NP hard |

Question 3 Explanation:

Question 4 |

Consider the decision problem 2CNFSAT defined as follows:

{ \phi | \phi is a satisfiable propositional formula in CNF with at most two literal per clause}

For example, \phi =(x_{1} \vee x_{2}) \wedge (x_{1} \vee \bar{x_{3}}) \wedge (x_{2} \vee x_{4}) is a Boolean formula and it is in 2CNFSAT.

The decision problem 2CNFSAT is

{ \phi | \phi is a satisfiable propositional formula in CNF with at most two literal per clause}

For example, \phi =(x_{1} \vee x_{2}) \wedge (x_{1} \vee \bar{x_{3}}) \wedge (x_{2} \vee x_{4}) is a Boolean formula and it is in 2CNFSAT.

The decision problem 2CNFSAT is

NP-Complete | |

solvable in polynomial time by reduction to directed graph reachability. | |

solvable in constant time since any input instance is satisfiable | |

NP-hard, but not NP-complete |

Question 4 Explanation:

Question 5 |

Suppose a polynomial time algorithm is discovered that correctly computes the largest clique
in a given graph. In this scenario, which one of the following represents the correct Venn
diagram of the complexity classes P, NP and NP Complete (NPC)?

A | |

B | |

C | |

D |

Question 5 Explanation:

There are 5 questions to complete.