An Introduction to Randomized algorithms - School of Technology ...

LP-based Randomized RoundingILP Formulation : Maximize ∑ 1≤j≤m z jsubject to : ∑ i∈C +jy i , z j ∈ {0, 1} for every i and j.y i + ∑ i∈C − (1 − y i ) ≥ z j ∀jjLP Relaxation : allow each y i , z j ∈ [0, 1].An optimal solution (y ∗i , z∗ j ) i,j of a LP can be foundefficiently.Rand. Rounding : Independently and randomly set eachy i = 1 with probability yi ∗ . Let g be the resulting assignment.|C j | = k ⇒ Pr(C j is satisfied by g) ≥ β k zj ∗ whereβ k = 1 − (1 − 1/k) k ≥ 1 − 1/e.E[#f ] ≥ ∑ ∑k j:|C j |=k β kzj∗ ≥ (1 − 1/e)OPT (ILP).