Approximation Algorithms Multiway Cut and k-Cut - Corelab

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Tight example • Consider a graph with 2k vertices, k of which form a k-cycle with edge weight equal to 1 and k terminals, each one connected to one of the vertices of the cycle with edges of weight 2-ε for a small ε>0. • The algorithm computes a solution of weight (k-1)(2-ε), while the optimal multiway cut has weight k. NTUA, Spring Semester 2009 4/9
Construction of a Gomory-Hu tree • Consider a tree T with one node, the set S 0 =V • Select a set S i , |S i |≥2 and select 2 vertices u,v of S i . • Compute a minimum u-v cut in G’, where G’ is the graph obtained by G and collapsing each subtree of S i into a single supernode. We compute the minimum cut between u and v in the new graph and obtain a partition Vn containing u and V 2 containing v. • Graph T is modified by breaking S i into two sets S i1 =S i IV 1 and S i2 =S i IV 2 . We add an edge between them with cost the cost just calculated. • We connect a subtree of T to S i1 if its supernode was in the same partition as u in the minimum cut, otherwise we connect it to S i2 . NTUA, Spring Semester 2009 5/9
Page 1 and 2: Approximation Algorithms Multiway C
Page 3: 2-2/k approximation algorithm for m
Page 7 and 8: (2-2/k) approximation algorithm •
Page 9: Thank you! NTUA, Spring Semester 20

Approximation Algorithms Multiway Cut and k-Cut - Corelab

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