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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 20: Primal-dual algorithm for MST: The Algorithm Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay We continue with the job of designing a primal dual algorithm for MSTs. Here is the LP formulation. min X e c e x e s.t. X e crosses ¥ x e j ¥ j 1 8 ¥ x e 8 e 2 E Here is the dual. max X ¥ y ¥ ( j ¥ j 1) s.t. X e crosses ¥ y ¥ c e 8 e 2 E y ¥ 8 ¥ In the algorithm, rst note that we need to maintain the y ¥ s. Initially they are all zeroes. At any stage we need to improve the cost function. In general we can do that by increasing some of the y s and decreasing some. To keep things as simple as possible, we will try and do this by only increasing one of them at a time. Let us assume that all c e s are strictly positive, so initially, none of the constraints are tight. We wish to increase the cost function. To do so we have to increase some y ¥ . Which one? It seems that we gain the most by increasing the one where each vertex is in a separate partition. So, suppose we start to increase this. How much can we increase it by? We see that we can increase this upto the weight of the minimum weight edge. At this point the inequalities corresponding to all edges of minimum weight will be tight....
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This note was uploaded on 01/04/2010 for the course CSE CS435 taught by Professor Profsundar during the Summer '09 term at IIT Bombay.
- Summer '09
- Computer Science