assignment4_sol - CSIS 0250B Design and Analysis of...

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CSIS 0250B Design and Analysis of Algorithms Assignment 4 Sampled Solution 1. As matching M of G with n - 1 edges are given, we have 1 vertex on each side of the bipartite graph that are not matched. We construct a directed graph G 0 from G . For each edge e ( u, v ) E - M , we create a directed edge e ( u, v ) on G 0 . For each edge e ( u, v ) M , we create a directed edge e ( v, u ). WLOG, let the unmatched vertex on the left hand side be s and the one on the right hand side be t . We apply BFS/DFS on G 0 starting with the vertex s and find a path from s to t . If such path exists, we conclude that there exists a perfect matching; otherwise we conclude there does not exist one. Correctness We try to augment the matching like the Maximum Flow Algo- rithm does. The G 0 is in fact a reconstruct of the residual network we would use to solve the bipartite problem on G . If there exists a perfect matching, we should be able to find an augmenting path that increases the current flow by 1, i.e. the path from s to t .
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