Lecture 12 Gadgets and NP.pdf - Comp 360 Algorithm Design(Fall 2017 Lecture 12 Reduction via “Gadgets” and Definition of NP Instructor Yang Cai

Lecture 12 Gadgets and NP.pdf - Comp 360 Algorithm...

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Comp 360: Algorithm Design (Fall 2017) Lecture 12: Reduction via “Gadgets” and Definition of NP Instructor: Yang Cai Basedon Kevin Wayne’sSlides.
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Review: Polynomial-time Reduction
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3 Polynomial-Time Reduction Desiderata. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? Reduction. Problem X polynomial reduces to problem Y if arbitrary instances of problem X can be solved using: n Polynomial number of standard computational steps, plus n Polynomial number of calls to oracle that solves problem Y. Notation. X P Y. We read this as “X is polynomial-time reducible to Y” or “Y is at least as hard as X”. Remarks. n We pay for time to write down instances sent to black box instances of Y must be of polynomial size. computational model supplemented by special piece of hardware that solves instances of Y in a single step
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4 Polynomial-Time Reduction Purpose. Classify problems according to relative difficulty. Design algorithms. If X P Y and Y can be solved in polynomial-time, then X can also be solved in polynomial time. Establish intractability. If X P Y and X cannot be solved in polynomial- time, then Y cannot be solved in polynomial time. Establish equivalence. If X P Y and Y P X, we use notation X P Y.
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5 Polynomial-Time Reduction Basic strategies. n Reduction by simple equivalence. n Reduction from special case to general case. n Reduction by encoding with gadgets.
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6 Literal: A Boolean variable or its negation.
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