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# lecture19 - CS 473 Algorithms Chandra Chekuri...

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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set Part I Reductions Continued Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set Polynomial Time Reduction A polynomial time reduction from a decision problem X to a decision problem Y is an algorithm A that has the following properties: given an instance I X of X , A produces an instance I Y of Y A runs in time polynomial in | I X | . This implies that | I Y | (size of I Y ) is polynomial in | I X | I X is a YES instance of X iff I Y is a YES instance of Y Notation: X ≤ P Y if X reduces to Y Proposition If X ≤ P Y then a polynomial time algorithm for Y implies a polynomial time algorithm for X. Such a reduction is called a Karp reduction. Most reductions we will need are Karp reductions Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set A More General Reduction Turing Reduction (the one given in the book) Problem X polynomial time reduces to Y if there is an algorithm A for X that has the following properties: on any given instance I X of X , A uses polynomial in | I X | “steps” a step is either a standard computation step or a sub-routine call to an algorithm that solves Y Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set A More General Reduction Turing Reduction (the one given in the book) Problem X polynomial time reduces to Y if there is an algorithm A for X that has the following properties: on any given instance I X of X , A uses polynomial in | I X | “steps” a step is either a standard computation step or a sub-routine call to an algorithm that solves Y Note: In making sub-routine call to algorithm to solve Y , A can only ask questions of size polynomial in | I X | . Why? Above reduction is called a Turing reduction. Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle. Goal Compute the maximum number of non-overlapping arcs. Reduced to the following problem:? Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle. Goal Compute the maximum number of non-overlapping arcs. Reduced to the following problem:? Input Collection of intervals on the line. Goal Compute the maximum number of non-overlapping intervals. How? Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3-SAT 3-SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle....
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lecture19 - CS 473 Algorithms Chandra Chekuri...

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