CS 294 : PCP and Hardness of Approximation January 23 , 2006
نویسنده
چکیده
We assume that all combinatorial objects that we refer to (graphs, boolean formulas, families of sets) are represented as binary strings. For a binary string x, we denote its length as |x|. We represent a decision problem as a language, that is, as the set of all inputs for which the answer is YES. We define P as the class of languages that can be decided in polynomial time. We define NP as the class of languages L such that there is a polynomial time computable predicate V and a polynomial q() such that x ∈ L if and only if there is w, |w| ≤ q(|x|) such that V (x,w) accepts. We think of w as a proof, or witness that x is in the language. For two languages L1 and L2, we say that L1 reduces to L2, and we write L1 ≤m L2 if there is polynomial time computable f such that x ∈ L1 if and only if f(x) ∈ L2. A language A is NP-hard if every language L in NP reduces to A. A language is NP-complete if it is NP-hard and it belongs to NP.
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