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Unformatted text preview: Massachusetts Institute of Technology Handout 18 6.854J/18.415J: Advanced Algorithms Fri, November 13, 2009 David Karger Problem Set 8 Solutions Fri, November 13, 2009 Problem 1. (a) Given sets S 1 ...S n , set cover asks for a minimum size collection C ⊆ { S 1 ...S n } such that for every s ∈ ∪ S i , exists an S i ∈ C such that s ∈ S i . Consider the following linear programming relaxation: min X i x S i subject to ∑ i : e ∈ S i x S i ≥ 1 ∀ e ∈ U x S i ≥ x S i ≤ 1 (b) Suppose that we include S i in the set cover with probability x S i . Consider an element e . Then we have: Pr( e is covered) = Pr( ∃ i,e ∈ S i ,S i is set to 1) Since we round include each S i in the set cover independently, we have: Pr( ∃ i,e ∈ S i ,S i is included) = 1 Pr( ∀ i,e ∈ S i S i is not included) The probability that S i is not included is 1 x S i , so we have: Pr( ∃ i,e ∈ S i ,S i is included) = 1 Y i : e ∈ S i (1 x S i ) Now, Q i : e ∈ S i (1 x S i ) is maximized when x S i = 1 { i : e ∈ S i } . Therefore, we have: Pr( ∃ i,e ∈ S i ,S i is included) = 1 Y i : e ∈ S i (1 x S i ) ≥ 1 1 1 { i : e ∈ S i } { i : e ∈ S i } ≥ 1 1 e ≥ 1 3 (c) We perform the above rounding procedure Θ(log n ) times. The set cover T we output is the union of all sets S i that are selected in any iteration of the rounding procedure. The probability that any particular element e is left uncovered after Θ(log n ) rounds is ( 1 3 ) Θ(log n ) , which is 1 n c . By the union bound, the probability that all elements are covered is at least 1 n n c . 2 Handout 18: Problem Set 8 Solutions In every rounding procedure, note that the expected cost of the rounding is exactly the optimum of the linear program. Therefore, in expectation over all of the rounding procedures, the cost of our rounding is O (log n ) times the optimum of the linear program, which is at most the optimum of the set cover problem. By using Markov’s inequality, we get that T covers all elements using at most O (log n )OPT number of sets with constant probability. By repeating this entire algorithm over n times, we get the result with high probability. (d) Suppose after solving the linear program, we multiply all the variables by c log n . We again apply the rounding procedure: with probability c log nx S i , we include S i in the set cover (if c log nx S i > 1, we just add S i to the set cover). Consider a given element e ∈ S , and consider the expected number of sets that cover e . In particular, let X i be the random variable that is 1 if S i is put in the cover, and 0 otherwise. The expected number of sets that cover a fixed element e is: Ex X i : e ∈ S i X i !...
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 Fall '09
 DavidKarger
 Linear Programming, Algorithms, Linear Programming Relaxation, Probability theory, Randomness

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