A Faster Algorithm for Minimum-Cost Bipartite Perfect Matching in Planar Graphs

Given a weighted planar bipartite graph G (A boolean OR B, E) where each edge has a positive integer edge cost, we give an (O) over tilde (n(4/3) log nC) time algorithm to compute minimum-cost perfect matching; here C is the maximum edge cost in the graph. The previous best known planarity exploiting algorithm has a running time of O (n(3/2) log n) and is achieved by using planar separators (Lipton and Tarjan '80). Our algorithm is based on the bit-scaling paradigm (Gabow and Tarjan '89). For each scale, our algorithm first executes O (n(1/3)) iterations of Gabow and Tarjan's algorithm in O (n(4/3)) time leaving only O (n(2/3)) vertices unmatched. Next, it constructs a compressed residual graph H with O (n(2/3)) vertices and O (n) edges. This is achieved by using an r-division of the planar graph G with r = n(2/3). For each partition of the r-division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortest-path data structures, the remaining O (n(2/3)) vertices are matched by iteratively computing a minimum-cost augmenting path each taking (O) over tilde (n(2/3)) time. Augmentation changes the residual graph, so the algorithm updates the compressed representation for each affected partition in (O) over tilde (n(2/3)) time. We bound the total number of affected partitions over all the augmenting paths by O (n(2/3) log n). Therefore, the total time taken by the algorithm is (O) over tilde (n(4/3)).