Advanced Approximation Algorithms ( CMU 18 - 854 B , Spring 2008 ) Lecture 27 : Algorithms for Sparsest Cut Apr 24 , 2008
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چکیده
In lecture 19, we saw an LP relaxation based algorithm to solve the sparsest cut problem with an approximation guarantee of O(logn). In this lecture, we will show that the integrality gap of the LP relaxation is O(logn) and hence this is the best approximation factor one can get via the LP relaxation. We will also start developing an SDP relaxation based algorithm which provides an O( √ log n) approximation for the uniform sparsest cut problem (where demands between all pairs of vertices is Dij = 1), and an O( √ logn log logn) algorithm for the sparsest cut problem with general demands.
منابع مشابه
Advanced Approximation Algorithms ( CMU 18 - 854 B , Spring 2008 ) Lecture 16 : Gaps for Max - Cut Mar 6 , 2008
Sdp(G) ≥ Opt(G) ≥ AlgGW (G) Also, we know the following as well, Theorem 1.1. [2] For all graphsG, AlgGW (G) ≥ αGWSdp(G), where αGW is a constant≈ 0.878. But this only gives us a ratio comparing Sdp(G) and AlgGW (G), and is thus a worry in the following sense: If the graph is such that OPT (G) ≈ 0.5, the Goemans Williamson algorithm could actually return a cut of size ≤ 0.5 (if Sdp(G) were also...
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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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