Lecture 16 : Maximum Cut , Integerality Gap and Metric Embedding
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چکیده
In the last lecture, we introduced the classic problem of maximum cut, where the objective is to find a cut of the given graph with maximum weight. We also saw some techniques give 1/2-approximation, such as random assignment and local search. In this lecture we will introduce the semidefinite-programming (SDP) relaxation for this problem. This approach, proposed by Goemans &Williamson in 90’s, improves upon 1/2 and is in fact the best we can do, under certain complexity theory hypothesis.
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Advanced Approximation Algorithms ( CMU 15 - 854 B , Spring 2008 ) Lecture 20 : Embeddings into Trees and L 1 Embeddings March 27 2008
where c(E(S, S̄)) is the sum of the weights of the edges that cross the cut, and D(S, S̄) is the sum of the demands of the pairs (si, ti) that are separated by the cut. We recall that optimizing over the set of cuts is equivalent to optimizing over `1 metrics, and is NP-hard. Instead, we may optimize over the set of all metrics. In this lecture, we bound the gap introduced by this relaxation by s...
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