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4. Assume that: • Problem Q₁ reduces to problem Q2 in polynomial time, and ● problem Q2 reduces to problem Q3 in polynomial time, and problems Q₁, Q2 and Q3 are all in NP. (a) If Q2 can be solved in polynomial time, what can be concluded about Q₁? (The answer may be nothing.) (b) If Q2 can be solved in polynomial time, what can be concluded about Q3? (The answer may be nothing.) (c) If Q2 is NP-complete, what can be concluded about Q₁? (The answer may be nothing.) (d) If Q2 is NP-complete, what can be concluded about Q3? (The answer may be nothing.)

4. Assume that: • Problem Q₁ reduces to problem Q2 in polynomial time, and ● problem Q2 reduces to problem Q3 in polynomial time, and problems Q₁, Q2 and Q3 are all in NP. (a) If Q2 can be solved in polynomial time, what can be concluded about Q₁? (The answer may be nothing.) (b) If Q2 can be solved in polynomial time, what can be concluded about Q3? (The answer may be nothing.) (c) If Q2 is NP-complete, what can be concluded about Q₁? (The answer may be nothing.) (d) If Q2 is NP-complete, what can be concluded about Q3? (The answer may be nothing.)

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4. Assume that: • Problem Q₁ reduces to problem Q2 in polynomial time, and • problem Q2 reduces to problem Q3 in polynomial time, and • problems Q1, Q2 and Q3 are all in NP. (a) If Q₂ can be solved in polynomial time, what can be concluded about Q₁? (The answer may be nothing.) (b) If Q₂ can be solved in polynomial time, what can be concluded about Q3? (The answer may be nothing.) (c) If Q2 is NP-complete, what can be concluded about Q₁? (The answer may be nothing.) (d) If Q2 is NP-complete, what can be concluded about Q3? (The answer may be nothing.)
Transcribed Image Text:4. Assume that: • Problem Q₁ reduces to problem Q2 in polynomial time, and • problem Q2 reduces to problem Q3 in polynomial time, and • problems Q1, Q2 and Q3 are all in NP. (a) If Q₂ can be solved in polynomial time, what can be concluded about Q₁? (The answer may be nothing.) (b) If Q₂ can be solved in polynomial time, what can be concluded about Q3? (The answer may be nothing.) (c) If Q2 is NP-complete, what can be concluded about Q₁? (The answer may be nothing.) (d) If Q2 is NP-complete, what can be concluded about Q3? (The answer may be nothing.)
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