lecture16 - CSci 5403 COMPLEXITY THEORY LECTURE XVI:...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XVI: COUNTING PROBLEMS AND RANDOMIZED REDUCTIONS 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. #SAT( φ ) = # satisfying assignments to φ . Definition. ƒ is #P-hard iff #P µ FP ƒ . Definition. ƒ is #P-complete iff #P µ FP ƒ and ƒ 2 #P. Theorem. (Shocking!) #SAT is #P-complete. Theorem. The following counting problems are parsimoniously equivalent: #PERFECT BPMATCH: Given: bipartite graph G = (U,V,E) where E µ U £ V Output: # of perfect matchings in G. #CYCLE COVER: Given: directed graph G Output: the number of cycle covers in G. MATRIX PERMANENT: Given: n £ n 0-1 matrix A Output: the permanent of A, π i n a i, π (i) . Theorem. PERMANENT is #P-Complete. Proof. We show that #3SAT µ FP #CYCLE COVER . The reduction has 3 gadget types: choice, clause, and XOR.
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CLAUSE GADGETS For each clause (x Ç y Ç z) include the subgraph: No cycle cover can include all three outer edges. For each subset of the outer edges, exactly one cycle cover includes that subset. x y z CHOICE GADGETS For each variable x i include the subgraph: There are exactly two cycle covers, one per outer edge. They correspond to the choices of x i . XOR GADGETS A clause gadget outer edge is connected to its choice gadget outer edge by an XOR gadget: An XOR gadget is a clever device that forces any cycle cover to use exactly one of the edges. If P NP, can we implement an XOR gadget? XOR GADGETS II A clause gadget outer edge is connected to its choice gadget outer edge by an XOR gadget: An XOR gadget is a clever device that multiplies the number of cycle covers using exactly one of the edges by 4, and the number using both edges by M. #SAT(
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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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lecture16 - CSci 5403 COMPLEXITY THEORY LECTURE XVI:...

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