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Unformatted text preview: Graph Partitioning Problems Lecture 18: March 14 s1 s3 s4 s2 T1 T4 T2 T3 s1 s4 s2 s3 t3 t1 t2 t4 A region R1 R2 C1 C2 Graph Partitioning Problems General setting: to remove a minimum (weight) set of edges to cut the graph into pieces. Examples: Minimum (st) cut Multiway cut Multicut Sparsest cut Minimum bisection Minimum st Cut s Mininium st cut = Max st flow Minimum st cut = minimum (weighted) set of edges to disconnect s and t t Multiway Cut Given a set of terminals S = {s1, s2, …, sk}, a multiway cut is a set of edges whose removal disconnects the terminals from each other. The multiway cut pr oblem asks for the minimum weight multiway cut. s1 s3 s4 s2 Multicut Given k sourcesink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}, a multicut is a set of edges whose removal disconnects each sourcesink pair. The multicut pr oblem asks for the minimum weight multicut. s1 s4 s2 s3 t3 t1 t2 t4 Multicut vs Multiway cut Given k sourcesink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}, a multicut is a set of edges whose removal disconnects each sourcesink pair. Given a set of terminals S = {s1, s2, …, sk}, a multiway cut is a set of edges whose removal disconnects the terminals from each other. What is the relationship between these two problems? Multicut is a generalization of multiway cut. Why? Because we can set each (si,sj) as a sourcesink pair. Sparsest Cut Given k sourcesink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}. For a set of edges U, let c(U) denote the total weight. Let dem(U) denote the number of pairs that U disconnects. The spar sest cut pr oblem asks for a set U which minimizes c(U)/ dem(U). I n other words, the sparsest cut problem asks for the most cost effective way to disconnect sourcesink pairs, i.e. the average cost to disconnect a pair is minimized. Sparsest Cut Suppose every pair is a sourcesink pair. For a set of edges U, let c(U) denote the total weight. Let dem(U) denote the number of pairs that U disconnects. The spar sest cut pr oblem asks for a set U which minimizes c(U)/ dem(U). S VS Minimize Sparsest Cut This is related to the normalized cut in image segmentation. Minimum Bisection The minimum bisection pr oblem is to divide the vertex set into two equal size parts and minimize the total weights of the edges in between. This problem is very useful in designing approximation algorithms for other problems – to use it in a divideandconquer strategy. Relations Minimum cut Multiway cut Multicut Sparsest cut Minimum bisection Results Minimum cut Polynomial time solvable....
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This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.
 Spring '09
 xulei

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