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Lecture23 - C&O 355 Lecture 23 N Harvey Topics...

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C&O 355 Lecture 23 N. Harvey
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Topics Weight-Splitting Method Shortest Paths Primal-Dual Interpretation Local-Ratio Method Max Cut
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k Weight-Splitting Method Let C ½ R n be set of feasible solutions to some optimization problem. Let w 2 R n be a “weight vector”. x is “optimal under w” if x optimizes min { w T y : y 2 C } Lemma: Suppose w = w 1 + w 2 . Suppose that x is optimal under w 1 , and x is optimal under w 2 . Then x is optimal under w. Hassin ‘82 Frank ‘81
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Weight-Splitting Method Appears in this paper: Andr á s Frank
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Weight-Splitting Method Scroll down a bit… Andr á s Frank Weight-Splitting Method was discovered in U. Waterloo C&O Department!
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ShortestPath ( G, S, t, w ) Input: Digraph G = (V,A), source vertices S μ V, destination vertex t 2 V, and integer lengths w(a), such that w(a)>0, unless both endpoints of a are in S. Output: A shortest path from some s 2 S to t. If t 2 S, return the empty path p=() Set w 1 (a)=1 for all a 2 ± + (S), and w 1 (a)=0 otherwise Set w 2 = w - w 1 . Set S’ = S [ { u : 9 s 2 S with w 2 ( (s,u) ) = 0 } Set p’ = (v 1 ,v 2 , ,t) = ShortestPath ( G, S’, t, w 2 ) If v 1 2 S, then set p=p’ Else, set p=(s,v 1 ,v 2 , ,t) where s 2 S and w 2 ( (s,v 1 ) )=0 Return path p To find shortest s-t path, run ShortestPath(G, {s}, t, w)
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Claim: Algorithm returns a shortest path from S to t. Proof: By induction on number of recursive calls. If t 2 S, then the empty path is trivially shortest. Otherwise, p’ is a shortest path from S’ to t under w 2 . So p is a shortest path from S to t under w 2 . (Note: length w 2 (p)=length w 2 (p ’), because if we added an arc, it has w 2 -length 0.) Note: p cannot re-enter S, otherwise a subpath of p would be a shorter S-t path. So p uses exactly one arc of ± + (S). Correctness of Algorithm s s' S t Path p This is a shorter S-t path
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Claim: Algorithm returns a shortest path from S to t. Proof: By induction on number of recursive calls. If t 2 S, then the empty path is trivially shortest. Otherwise, p’ is a shortest path from S’ to t under w 2 . So p is a shortest path from S to t under w 2 . (Note: length w 2 (p)=length w 2 (p’), because if we added an arc, it has w 2 -length 0.) Note: p cannot re-enter S, otherwise a subpath of p would be a shorter S-t path. So p uses exactly one arc of ± + (S). So length w 1 (p)=1. But any S-t path has length at least 1 under w 1 . So p is a shortest path from S to t under w 1 . ) p is a shortest S-t path under arc-lengths w, by the Weight-Splitting Lemma. ¥ Correctness of Algorithm
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Another IP & LP for Shortest Paths Make variable x a for each arc a 2 A The IP is: Corresponding LP & its dual: Make variable y C for each S-t cut C Theorem: The Weight-Splitting Algorithm finds optimal primal and dual solutions to these LPs.
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