Set cover problem: definition - Organic Origami

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Set cover (I)LP min ∑ s∈S c(S)x S s.t. ∑ x S ≥ 1, ∀e ∈ U S: e∈S x S ∈ {0, 1} ∀S ∈ S To get a polynomially solvable relaxation, replace the 0-1 constraints by 0 ≤ x S ≤ 1 for all S.
Set cover rounding How to round? Deterministically, it’s not easy to get a good bound. However, randomized rounding here is very simple. Set cover randomized rounding procedure: Suppose xS ∗ for every S are the values of x S in the optimal LP solution. For each S independently, flip a coin biased by xS ∗ , so that S is included in the cover with probability exactly xS ∗. Write x S for the rounded value of x S . Easy: the expected cost of this rounding procedure is ∑ c(S) P[x S = 1] = ∑ c(S)xS ∗ = OPT LP ≤ OPT . S S How about the success probability?
Page 1 and 2: Set cover problem: definition Given
Page 3 and 4: Set cover greedy: analysis Suppose
Page 5 and 6: Is the analysis tight? There’s an
Page 7: A completely different approach: li
Page 11: Set cover rounding: success probabi

Set cover problem: definition - Organic Origami

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