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

Mudabir Kabir Asathulla, Sanjeev Khanna, Nathaniel Lahn, Sharath Raghvendra

Abstract: Given a weighted planar bipartite graph G(A ∪ B, E) where each edge has an integer edge cost, we give an Õ(n4/3log 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(n3/2log 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(n1/3) iterations of Gabow and Tarjan’s algorithm in O(n4/3) time leaving only O(n2/3) vertices unmatched. Next, it constructs a compressed residual graph H with O(n2/3) vertices and O(n) edges. This is achieved by using an r-division of the planar graph G with r=n2/3. 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(n2/3) vertices are matched by iteratively computing a minimum-cost augmenting path, each taking Õ(n2/3) time. Augmentation changes the residual graph, so the algorithm updates the compressed representation for each partition affected by the change in Õ(n2/3) time. We bound the total number of affected partitions over all the augmenting paths by O(n2/3 log n). Therefore, the total time taken by the algorithm is Õ(n4/3).