19_TheHungarianAlgorithm - Thursday Dr Daniel Hughes...

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CSC 30155 Thursday 18/11/10 Dr. Daniel Hughes [email protected]
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Today’s Lecture The Hungarian Algorithm (40 mins) Feedback Response (10 mins)
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CSC 30155 The Hungarian Algorithm Dr. Daniel Hughes [email protected]
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Bipartite Graphs 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 . V 1 = {1, 2, 3}. V 2 = {4, 5, 6, 7}.
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Maximum Matching Using Alternating Paths 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).
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Maximum Matching Using Alternating Paths 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). An alternating path joining two uncovered vertices is called an M- augmenting path (see the red path in the graph in the middle).
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Berge’s Result (1957) So we could find a maximum matching by generating and testing all alternating paths and using them to augment the matching This does not give us an efficient algorithm for finding a maximum matching.
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