CS G 399 : Algorithmic Power Tools I Scribe : Bishal Thapa

CS G 399 : Algorithmic Power Tools I Scribe

Where Z∗ is OPT, we know Z∗ = 3x1 + 2x ∗ 2 + 8x ∗ 3, for some x ∗ 1, x ∗ 2, x ∗ 3 ∈ P . By adding two of the inequalities, we arrive at 2x1 + x2 + x3 ≥ 15. Since x1, x2, x3 ≥ 0, we know that Z∗ ≥ 15. But we aren’t limited to addition, multiplication is another way the equations can be combined. So how is this new formulation bouned? This is done by using the dual formulation, D of the minimizat...

CS 7880 : Algorithmic Power Tools Scribe : Eric

As with the tail bounds we saw in the last lecture, Chernoff’s formula is used to bound the probability that a random variable deviates from its mean. For this bound, however, the random variable under consideration will have a very specific structure. Let X1, . . . , Xn be independent {0, 1}-valued random variables, and define pi = Pr[Xi = 1] = E[Xi]. Then, X = ∑ iXi will be the random variabl...

CS G399: Algorithmic Power Tools I Scribe: Jian Wen Lecture Outline:

The k-median problem is a variant of the uncapacitated facility location problem that we covered last week, with the difference that instead of a cost for each facility, we have a bound k on the number of facilities. Here is the definition of k-median problem: Problem 1. Given a set of locations(clients) V , a set F̃ of facilities, a cost function between two locations C : (V ⋃ F̃ ) ∗ (V ⋃ F̃ ) → ...

2 Randomized rounding approach for UFL

CS 7880 : Algorithmic Power Tools Scribe

• Two Simple Applications of the LLL • Packet Routing In this lecture, we prove the general form of the Lovász Local Lemma, and see two applications of the lemma: hypergraph coloring and finding satisfying assignments for Boolean CNF formulas. We then begin studying packet routing, another area in which we will apply the lemma. In the last lecture, we proved the symmetric form of the Lovász Loc...