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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, UrbanaChampaign Fall 2009 Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3SAT 3SAT and Independent Set Part I Reductions Continued Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3SAT 3SAT 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 3SAT 3SAT 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 subroutine call to an algorithm that solves Y Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3SAT 3SAT 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 subroutine call to an algorithm that solves Y Note: In making subroutine 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 3SAT 3SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle. Goal Compute the maximum number of nonoverlapping arcs. Reduced to the following problem:? Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3SAT 3SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle. Goal Compute the maximum number of nonoverlapping arcs. Reduced to the following problem:? Input Collection of intervals on the line. Goal Compute the maximum number of nonoverlapping intervals. How? Chekuri CS473ug The Satisfiability Problem (SAT) Sat and 3SAT 3SAT and Independent Set Example of Turing Reduction Recall home work problem: Input Collection of arcs on a circle....
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 Fall '08
 Chekuri,C
 Algorithms, Computational complexity theory, independent set, Boolean satisfiability problem, chekuri, CS473ug

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