lec20-handout.pdf

# lec20-handout.pdf - Announcements CMPSCI 311 Introduction...

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CMPSCI 311: Introduction to Algorithms Lecture 20: Reductions and Intractability Akshay Krishnamurthy University of Massachusetts Last Compiled: April 23, 2018 Announcements Quiz due tonight HW 6 due 5/1 (Tuesday night!), and extra credit Midterms back on wednesday (Solutions up tonight) Last discussion on friday Final Exam: Friday 5/4, 3:30-5:30pm, Marcus Hall 131. Recap Problem X is a set of strings s , the YES instances. Algorithm A solves X if A ( s ) = true iff s X . B is polytime certifier for X if B is polytime algorithm of two inputs s and t (a hint). s X iff exists t with | t | ≤ p ( | s | ) and B ( s, t ) = True . P – class of problems with polytime algorithm. NP – class of problems with polytime certifier. X is NP-Complete iff Y P X for all Y ∈ NP . Example Problem ( X ) IndependentSet Instance ( s ) Graph G and number k Algorithm ( A ) Try all subsets and check ( Runtime? ) Hint ( t ) Which nodes are in the answer? Certifier ( B ) Are those nodes independent and size k ? Plan for today Review 3-SAT P CircuitSat HamCycle TSP Back to 3-SAT Claim. If Y is NP-complete and Y P X , then X is NP-complete. Theorem. 3-SAT is NP-Complete. Clearly in NP . Prove by reduction from CircuitSAT . Example. i1 i2 1 ¬ o

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The Reduction One variable x v per circuit node v .
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• Fall '09
• Computational complexity theory, NP-complete problems, Travelling salesman problem, NP-complete, Hamiltonian path problem

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