CS 7880 : Algorithmic Power Tools Scribe
نویسنده
چکیده
• 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 Local Lemma (LLL). Today we will prove the general form. Recall that the LLL is used to show that, for a set of " bad " events over a sample space, there exists a point in the sample space outside of all of the events, under certain conditions on the dependence and probabilities of the events.
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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...
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In this lecture, we first complete the analysis of “Greedy” approach we introduced in the last lecture, that gives a constant (6)-approximation polynomial-time solution for the Uncapacitated Facility Location problem. Then, we present a primal-dual approach of solving the problem, that yields a better approximation solution of cost within a factor of 3 of the optimal. The latter has a polynomia...
متن کاملCS 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 ) → ...
متن کاملNortheastern University 15 September 2009 CS 7880 : Algorithmic Power Tools Scribe
• Probabilistic method and applications • " Tail Bounds " • Chernoff bounds In this lecture, we complete the introduction to approximation algorithms and by using the max-cut and dominating set problem, we introduce the probabilistic method of solving problems. We also study some techniques for bounding the probability that a random variable takes value far from its expectation, known as tail b...
متن کاملCS 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...
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