lecture15 - So far we have seen problems of canonical form:...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XV: COUNTING PROBLEMS So far we have seen problems of canonical form: φ (x)? 9 x. φ (x), φ 2 2cnf C(x)? 9 x. φ (x) 9 x 8 y φ (x,y) 9 x 1 8 x 2 … Q n x n φ (x 1 …x n ). L NL P NP PH PSPACE Next two lectures: How many x satisfy φ (x)? #P PRONUNCIATION KEY (Courtesy Steven Rudich) #P = number-P (makes sense) = pound-P The British way = sharp-P The cool way. Definition. #P is the class of functions ƒ such that there exists a polytime TM M where ƒ(x) = |{ y : M(x,y) accepts }| Example. ƒ( φ ) = # of satisfying assignments to φ . The function ƒ is also called #SAT. Proposition. ƒ 2 #P. Proof. Let M( φ ,x) = accept iff φ (x) is true.
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2 #HAM: Given digraph G, how many Hamiltonian cycles does it have? #3COLOR: How many 3-colorings does G have? #TRIANGLE: How many triangles does G have? GRAPH RELIABILITY: How many subgraphs of G have a path from 1 to n? #CYCLE: How many simple cycles does G have? Examples… NP µ P #HAM NP µ P #3COLOR #TRIANGLE 2 FP Theorem. P #HAM µ P #CYCLE . Proof. We show how to compute #HAM(G) given an oracle for #CYCLE. Given G, produce the graph G’, where every edge (u,v) in G is replaced by the following subgraph: u v uv 11 uv 12 uv 21 uv 22 uv 31 uv 32 uv 41 uv 42 uv m1 uv m2 Every cycle of length in G corresponds to 2 m cycles in G’. Thus, if we let a
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This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.

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lecture15 - So far we have seen problems of canonical form:...

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