lecture22 - CS 435 : Linear Optimization Fall 2008 Lecture...

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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 22: Primal-Dual Algorithm for Shortest Paths Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay In this lecture we will design an algorithm for the classical shortest path problem. Input: A directed graph with positive integer weights w e on edges; two special vertices, s; t 2 V . Output: A shortest path from s to t . Our rst step is to write an ILP. We choose one variable per edge, x e . If x e is picked, x e = 1 else x e = 0. The cost function is min P u;v x uv w uv We will assume that there are no edges entering s or leaving t . The Constraints: 1. P u x su = 1 (The number of edges leaving s is 1) 2. P v x vt = 1 (The number of edges entering t is 1) 3. 8 p 2 V f s; t g P q x pq P r x rp = 0 (For all other vertices, the number of edges leaving them equals the number of edges entering them.) 4. For every edge uv , 0 x uv 1 ; x integral. We drop the integrality constraint and the upper bound constraints on x as in the case of MST and write the dual. Writing the dual is a little bit tricky and you have to be careful. The resulting dual willwrite the dual....
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lecture22 - CS 435 : Linear Optimization Fall 2008 Lecture...

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