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# L18-cut - Gr aph Par ti ti oni ng Pr obl ems T2 s1 T1 s2 T3...

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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

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Graph Partitioning Problems General setting: to remove a minimum (weight) set of edges to cut the graph into pieces. Examples: Minimum (s-t) cut Multiway cut Multicut Sparsest cut Minimum bisection
Minimum s-t Cut Mininium s-t cut = Max s-t flow Minimum s-t cut = minimum (weighted) set of edges to disconnect s and t

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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 problem asks for the minimum weight multiway cut. s1 s3 s4 s2
Multicut Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}, a multicut is a set of edges whose removal disconnects each source-sink pair. The multicut problem asks for the minimum weight multicut. s1 s4 s2 s3 t3 t1 t2 t4

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Multicut vs Multiway cut Given k source-sink pairs {(s1,t1), (s2,t2), ...,(sk,tk)}, a multicut is a set of edges whose removal disconnects each source-sink 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 source-sink pair.
Sparsest Cut Given k source-sink 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 sparsest cut problem asks for a set U which minimizes c(U)/ dem(U). In other words, the sparsest cut problem asks for the most cost effective way to disconnect source-sink pairs, i.e. the average cost to disconnect a pair is minimized.

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Sparsest Cut Suppose every pair is a source-sink 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 sparsest cut problem asks for a set U which minimizes c(U)/ dem(U). S V-S Minimize
Sparsest Cut This is related to the normalized cut in image segmentation.

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Minimum Bisection The minimum bisection problem 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 divide-and-conquer strategy.
Relations Minimum cut Multiway cut Multicut Sparsest cut Minimum bisection

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Results Minimum cut Polynomial time solvable.
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