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# Theory of Computation CS 3102

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This 4 page Class Notes was uploaded by Mrs. Carolyne Abbott on Monday September 21, 2015. The Class Notes belongs to CS 3102 at University of Virginia taught by David Evans in Fall. Since its upload, it has received 43 views. For similar materials see /class/209649/cs-3102-university-of-virginia in ComputerScienence at University of Virginia.

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Date Created: 09/21/15

NP Cornpleteness Lecture for CS 302 l Non deterministic polynomial time I Deterministic Polynomial Time The TM takes at most OnG steps to accept a string of length n I Nondeterministic Polynomial Time The TM takes at most Onc steps on each computation path to accept a string of length n P vs NP I Are nondeterministic Turing machines really more powerful efficient than deterministic ones I Essence of P vs NP problem l Traveling Salesperson Problem I You have to visit If cities I You want to make the shortest trip I How could you do this I What if you had a machine that could guess l The Class P and the Class NP I P L L is accepted by a deterministic Turing Machine in polynomial time I NP L L is accepted by a non deterministic Turing Machine in polynomial time I They are sets of languages Does Non Dete rminism matte r Firms Auwma av Push Down Automate Nol Yesl DFANFA DFA not NFA PNPP I No one knows it this is true I How can we make progress on this problem Progress I P NP it every NP problem has a deterministic polynomial algorithm I We could find an algorithm for every NP problem I Seems hard I We could use polynomial time reductions to find the hardest problems and just work on those Reductions I Real world examples a Finding your way around the city reduces to reading a map a Traveling from Richmond to Cville reduces to driving a car in Other suggestions i Polynomial tilne reductions I PARTITION nn2 nk we can split the integers into two sets which sum to halt I SUBSETSUM ltnn2 nKmgt there exists a subset which sums to m I 1 It I can solve SUBSETSUM how can I use that to solve an instance of PARTITION I 2 It I can solve PARTITION how can I use that to solve an instance of SUBSETSUM i Polynomial Reductions I 1 Partition REDUCES to SubsetSum I 2 SubsetSum REDUCES to Partition I Therefore they are equivalently hard in Partition ltp SubsetSum u SubsetSum ltp Partition I How long does the reduction take I How could you take advantage of an exponential time reduction NP Completeness I How would you define NPComplete I They are the hardest problems in NP i De nition ofNP Complete I Q is an NPComplete problem it I 1 Q is in NP I 2 every other NP problem polynomial time reducible to Q i Getting Started I How do you show that EVERY NP problem reduces to Q I One way would be to already have an NP Complete problem and just reduce from that kt My e at ry P3 NPrCornpleie 4 em Ptv Probl i Reminder Undecidability I How do you show a language is undecidable I One way would be to already have an undecidable problem and just reduce from that Li L2 Hailing La Problem 439 O H i SAT I SAT t t is a Boolean Formula with a satisfying assignment nquot 11 V m V 12 va V 112 V 13952 I Is SAT in NP I Cook Levin Theorem 1971 I SAT is NPComplete ly u want to see the prooI It is Theorem 7 37 m Sipser assigned readingl oryou can take heory on are not responsible lor Knowing the proof NP Complete I To prove a problem is NPComplete show a polynomial time reduction from 3SAT I Other NPComplete Problems 1 PARTITION a SUBSETSUM u CLIQUE u HAMILTONIAN PATH TSP u GRAPH COLORING u MINESWEEPER and many more I 3 SAT I 3SAT fl is in Conjunctive Normal Form each clause has exactly 3 literals and is satistiable I 3SAT is NPComplete 2SAT is in P I NP Completeness Proof Method I To show that Q is NPComplete I 1 Show that Q is in NP I 2 Pick an instance R of yourtavorite NP Complete problem ex 3 in 3SAT I 3 Show a polynomial algorithm to transform R into an instance of Q I Example Clique I CLIQUE ltGkgt G is a graph with a clique of size k I A clique is a subset of vertices that are all connected I Why is CLIQUE in NP I Reduce 3 SAT to Clique I Pick an instance of 3SAT 3 with kclauses I Make a vertex for each literal I Connect each vertex to the literals in other clauses that are not the negation I Any kclique in this graph corresponds to a satisfying assignment

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