19_TheHungarianAlgorithm

19_TheHungarianAlgorithm - CSC 30155 Thursday 18/11/10 Dr....

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Unformatted text preview: CSC 30155 Thursday 18/11/10 Dr. Daniel Hughes daniel.hughes@xjtlu.edu.cn Todays Lecture l The Hungarian Algorithm (40 mins) l Feedback Response (10 mins) CSC 30155 The Hungarian Algorithm Dr. Daniel Hughes daniel.hughes@xjtlu.edu.cn Bipartite Graphs l A bipartite graph is a graph in which the vertices can be partitioned into two sets, V 1 and V 2 and all edges go between V 1 and V 2 . l V 1 = {1, 2, 3}. V 2 = {4, 5, 6, 7}. Maximum Matching Using Alternating Paths l Let G be a graph and let M be any matching in G (see figure to the left). A path P = v 1 , , v m is said to be an alternating path with respect to M or an M- alternating path if (v i , v i+1 ) M if and only if (v i+1 , v i+2 ) M for 1 i m - 2 (see the green and red paths). Maximum Matching Using Alternating Paths l If G contains an M-alternating path joining two uncovered vertices then M cannot be a maximum matching the matching obtained by removing the edges in P M and adding the edges in P - M is larger (see figure below). l An alternating path joining two uncovered vertices is called an M- augmenting path (see the red path in the graph in the middle). Berges Result (1957) l So we could find a maximum matching by generating and testing all alternating paths and using them to augment the matching l This does not give us an efficient algorithm for finding a maximum matching. A Note on Data Structures...
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19_TheHungarianAlgorithm - CSC 30155 Thursday 18/11/10 Dr....

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