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Lecture22

# Lecture22 - C&O 355 Lecture 22 N Harvey Topics Integral...

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C&O 355 Lecture 22 N. Harvey

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Topics Integral Polyhedra Minimum s-t Cuts via Ellipsoid Method Weight-Splitting Method Shortest Paths
Minimum s-t Cuts (LP) Theorem: Every optimal BFS of (LP) is optimal for (IP). (IP) So to solve (IP), we can just solve (LP) and return an optimal BFS. To solve (LP), the separation oracle is: (Lecture 12) Check if y a < 0 for any a 2 A. If so, the constraint “y a ¸ 0” is violated. Check if dist y (s,t)<1. If so, let p be an s-t path with length y (p)<1. Then the constraint for path p is violated. So to compute min s-t cuts, we just need an algorithm to compute shortest dipaths! (Lecture 21)

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Shortest Paths in a Digraph Let G=(V,A) be a directed graph. Every arc a has a “length” w a >0. Given two vertices s and t, find a path from s to t of minimum total length. 3 1 8 7 1 5 1 2 2 2 3 2 4 1 2 s t These edges form a shortest s-t path
Shortest Paths in a Digraph Let b be vector with b s =1, b t =-1, b v =0 8 v 2 V n {s,t} Consider the IP: And the LP relaxation: Theorem: Every optimal BFS of (LP) is optimal for (IP).

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Lecture22 - C&O 355 Lecture 22 N Harvey Topics Integral...

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