Lecture 6

# To show is np hard it suces to nd a transformation t

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Unformatted text preview: time using a non-deterministic oracle (i.e., you can verify its guesses in polytime). • Thm: 3-SAT is NP-complete. • To show Π is NP-hard, it suﬃces to ﬁnd a transformation T ∈ P from another NP-hard problem Π (e.g., 3-SAT) since • Defn: Π is NP-hard if ∀Π ∈ N P. ∃T ∈ P. Π → Π. • Defn: Π is NP-complete if it is NP-hard and in NP. • To show Π is NP-complete, show it is NP-hard and that you can check (oracular) guesses in poly-time. • Conjecture: P = N P NP−hard NP NP−hard NP P P NP complete Exact inference in discrete Bayes nets is NP-complete • Thm: the decision problem “Is PB (Xi = x) > 0?” is NP-complete. • Proof. To show in NP: Given an assignment X1:n, we can check if Xi = x and then check if P (X1:n) > 0 in poly-time. To show NP-hard: we can encode any 3SAT problem as a polynomially sized Bayes net, as shown below. • P (X = 1|q1:n) > 0 iﬀ q1:n is a satisfying assignment. 4 ¥ ¤ 4 ¢ 4 £ 4 ¡ 4 T T NP → Π → Π T NP complete Complexity of approximate inference • Defn: An estimate ρ has absolute error |P (y |e) − ρ| ≤ . for P (y |e) if • Defn: An estimate ρ has relative error for P (y |e) if ρ ≤ P (y |e) ≤ ρ(1 + ) 1+ • Thm: Computing P (Xi = x) with re...
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## This document was uploaded on 03/28/2014 for the course CS 532 at UBC.

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