lecture18v - CS 435 : Linear Optimization Fall 2008 Lecture...

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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 18: An Algorithm for Matching Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay 1 Recap of previous lecture To make the discussion of the previous lecture precise we make a few de nitions. Fix a matching M in a graph G . Definition 1 A path in G will be called alternating (w.r.t. M ) if its edges alternate between those in M and those not in M . Definition 2 A vertex in G will be called unmatched or free if there are no edges from M incident on it. Definition 3 An augmenting path is an alternating path that starts from and ends on free (unmatched) vertices. Here is a formal theorem based on discussions done in the last lecture. Theorem 1 A matching M is maximum if and only if the resulting graph does not contain any aug- menting path. Make sure that you can prove this theorem. One direction is easy. If there is an augmenting path then the matching is not maximum. For the other direction assume that M is not maximum. Why should an augmenting path exist? Start your proof this way: Let M be an optimum matching. Algorithm for maximum matching: We start with a single edge in the matching M . In each iteration we rst nd an augmenting path P . We then exchange the matched and unmatched edges in....
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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.

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lecture18v - CS 435 : Linear Optimization Fall 2008 Lecture...

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