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lecture26

# lecture26 - Bounds on circuit computation so far CSci 5403...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XXVI: CIRCUIT LOWER BOUNDS “Bounds” on circuit computation so far: Size Hierarchy Theorem: For every ƒ(n) = o(2 n /n) there exists a function with complexity ω (ƒ(n)) Shannon bounds: “most” functions have complexity Θ (2 n /n) Relationships to uniform classes ( ˲ k, PH languages in SIZE( Θ (n k )), X ˧ P/poly ˰ Amazing(X), BPP ˧ P/poly, NC 1 ˧ L ˧ NL ˧ NC 2 …) This week: concrete bounds, specific languages. Theorem. (Razborov) CLIQUE k,n requires monotone circuits of size n Ω ( k) for all k n 1/4 . Exponential lower bounds! For a NP-hard function! Unfortunately, Theorem. (Razborov) Perfect matching requires superpolynomial monotone circuits. Definition. A monotone circuit contains only AND and OR gates. Definition. A boolean function ƒ is monotone iff ˲ x ˧ y, ƒ(x) ƒ(y). Definition. CLIQUE k,n (G) = 1 iff ʪ G,k ʫ ˥ CLIQUE Let S ˧ {1, …, n}. Then we let: C S (G) = ˭ i,j ˥ S e ij = 1, if S is a clique in G 0, otherwise we call C S a clique indicator. Proof idea: we will show that ˳ c c ,c such that (a) An OR of n c k clique indicators cannot compute CLIQUE n,k . (b) Every monotone circuit of size n c k can be simulated by an OR of n c k clique indicators. Thus, no monotone circuit of size O(n c k ) can compute CLIQUE n,k .

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2 We will make use of the following distributions on n-vertex graphs. Y
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lecture26 - Bounds on circuit computation so far CSci 5403...

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