Chis problem walks you through another example of the use of randomiz.pdf
nis problem walks you through another example of the use of randomize.pdf
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nis problem walks you through another example of the use of randomized rounding. onsider the Set-Cover problem: put: Subsets S1,S2,,Sm{1,2,,,n} utput: the smallest set S such that SSk= for all 1km. ( S does not have to be one of the Si.) (a) Write an instance of ILP that is equivalent to Set-Cover. Hint: set a variable x1{0,1} for each 1in. Your objective function should be a min and there should be m constraint, one for each set Si. b) As in class, consider the corresponding LP relaxation. Let {xi}i=1n be an optimal of the relaxation. Define xi={1,0,withprobabilityxi.else(i.e.:withprobability1xi). and set S={ixi=1}. Compute P(SSj=). Briefly justify your answer. (c) Conclude that P(SSj=)11/e. You can use the Arithmetic Mean- Geometric Mean inequality as well as others we used in class without prove. 'd) Consider the following algorithm: sample S like in part (b) and take the union until you cover all sets. Show that it is enough to do O(ln(m)) rounds in expectation. Hint: First, use (c) to conclude that after ln(m) rounds the probability of any Si is less than 1/m. Now take X= number of uncovered sets and conclude, using linearity of expectation, that E(X)<1 (e) (not graded) This algorithm yields a ln(n)-Approximation algorithm for Set-Cover. To see this, let Y be the size of the final set and Yk the set of the outcome of each round. Then E(Y)k=1mE(Yk)=ln(m)xiln(m)S where S is the optimal set cover. The last inequality uses the key observation that the optimal for the LP relaxation is a lower bound (you are minimizing!) for the optimal of the corresponding ILP, which is equivalent to Set-Cover..
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nis problem walks you through another example of the use of randomize.pdf
- nis problem walks you through another example of the use of randomized rounding. onsider the Set-Cover problem: put: Subsets S1,S2,,Sm{1,2,,,n} utput: the smallest set S such that SSk= for all 1km. ( S does not have to be one of the Si.) (a) Write an instance of ILP that is equivalent to Set-Cover. Hint: set a variable x1{0,1} for each 1in. Your objective function should be a min and there should be m constraint, one for each set Si. b) As in class, consider the corresponding LP relaxation. Let {xi}i=1n be an optimal of the relaxation. Define xi={1,0,withprobabilityxi.else(i.e.:withprobability1xi). and set S={ixi=1}. Compute P(SSj=). Briefly justify your answer. (c) Conclude that P(SSj=)11/e. You can use the Arithmetic Mean- Geometric Mean inequality as well as others we used in class without prove. 'd) Consider the following algorithm: sample S like in part (b) and take the union until you cover all sets. Show that it is enough to do O(ln(m)) rounds in expectation. Hint: First, use (c) to conclude that after ln(m) rounds the probability of any Si is less than 1/m. Now take X= number of uncovered sets and conclude, using linearity of expectation, that E(X)<1 (e) (not graded) This algorithm yields a ln(n)-Approximation algorithm for Set-Cover. To see this, let Y be the size of the final set and Yk the set of the outcome of each round. Then E(Y)k=1mE(Yk)=ln(m)xiln(m)S where S is the optimal set cover. The last inequality uses the key observation that the optimal for the LP relaxation is a lower bound (you are minimizing!) for the optimal of the corresponding ILP, which is equivalent to Set-Cover.