22_Reductions - CSC 30155 Polynomial Time Reductions Dr Ka...

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Unformatted text preview: CSC 30155 Polynomial Time Reductions Dr. Ka Lok Man On Behalf of Dr. Daniel Hughes [email protected] Polynomial-time Reducibility l Another important idea in the theory of NP-completeness is the idea of transforming one problem instance into another problem instance. We formalize this idea below. l We say that a problem L , defining some decision problem, is polynomial-time reducible to a problem M , if: there is a function f , computable in polynomial time, that takes an instance s of the problem L , and transforms it into an instance f(s) of the problem M such that s has answer “yes” if and only if f(s) has answer “yes”. l In other words, we can transform an input for one decision problem into an appropriate input for another decision problem . Furthermore, the first problem has a “yes” solution if and only if the second decision problem has a “yes” solution. l We use notation L poly M to signify that problem L is polynomial time reducible to problem M . NP-hardness l We say that a problem M , defining some decision problem, is NP-hard if every other problem L in NP is polynomial-time reducible to M . So this means that: l M is NP-hard if for every L ∈ NP , L poly M . l If a language M is NP-hard and it belongs to NP itself, then M is NP-complete . l NP-complete problems are some of the hardest problems in NP, as far as polynomial reducibility is concerned. Example: Cook-Levin Theorem l The circuit sat problem tests a given Boolean circuit for satisfiability, accepting it only if it is satisfiable. l The Cook-Levin Theorem states that Circuit-SAT is NP-complete. l Proof idea: The computation steps of any (reasonable) algorithm can be simulated by layers in appropriately constructed (polynomial time and size) Boolean Circuit. Other NP-complete problems l We have just noted that there is at least one NP- complete problem. l Using polynomial time reducibility we can show existence of other NP-complete problems. l The following useful result also helps to prove NP- completeness in many cases: Other NP-complete problems l Suppose that we have a new problem X and we think that X is NP-complete . How can we do this? l First, show that X ∈ NP , i.e. show that X has a polynomial-time nondeterministic algorithm, or equivalently, show that X has an efficient certifier. l Secondly, take a known NP-complete problem Y , and demonstrate a polynomial time reduction from Y to X , i.e. show that Y poly X . Types of Reduction l Let M be an NP-complete problem. There are three types of reduction: l Restriction: Noting that a known NP-complete problem is a special case of our problem L. l Local replacement: Dividing instances of M and L into basic units, and then showing how each basic unit of M can be mapped to a basic unit of L....
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This note was uploaded on 05/22/2011 for the course CSC 30155 taught by Professor Garyli during the Spring '11 term at University of Liverpool.

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22_Reductions - CSC 30155 Polynomial Time Reductions Dr Ka...

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