Mudabir Kabir Asathulla
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).
- Therese Biedl. 2001. Linear reductions of maximum matching. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA’01). SIAM, 825--826.Google Scholar
- Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum, and Christian Wulff-Nilsen. 2017. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. SIAM J. Comput. 46, 4 (2017), 1280--1303. DOI:https://doi.org/10.1137/15M1042929.Google ScholarDigital Library
- Michael B. Cohen, Aleksander Madry, Piotr Sankowski, and Adrian Vladu. 2017. Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m10/7 log W) time. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA’17). SIAM, 752--771. DOI:https://doi.org/10.1137/1.9781611974782.48Google ScholarCross Ref
- Sabine Cornelsen, Andreas Karrenbauer, and Shujun Li. 2012. Leveling the grid. In Proceedings of the Workshop on Algorithm Engineering and Experimentation (ALENEX’12). SIAM, 45--54. DOI:https://doi.org/10.1137/1.9781611972924.4Google ScholarCross Ref
- Jittat Fakcharoenphol and Satish Rao. 2006. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci. 72, 5 (2006), 868--889. DOI:https://doi.org/10.1016/j.jcss.2005.05.007Google ScholarDigital Library
- L. R. Ford and D. R. Fulkerson. 1957. A simple algorithm for finding maximal network flows and an application to the Hitchcock problem. Canadian J. Math. 9 (1957), 210--218. DOI:https://doi.org/10.4153/CJM-1957-024-0Google ScholarCross Ref
- H. N. Gabow and R. E. Tarjan. 1989. Faster scaling algorithms for network problems. SIAM J. Comput. 18 (Oct. 1989), 1013--1036. Issue 5. DOI:https://doi.org/10.1137/0218069Google Scholar
- Pawel Gawrychowski and Adam Karczmarz. 2018. Improved bounds for shortest paths in dense distance graphs. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP’18), Vol. 107. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 61:1--61:15. DOI:https://doi.org/10.4230/LIPIcs.ICALP.2018.61Google Scholar
- J. Hopcroft and R. Karp. 1973. An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 4 (1973), 225--231. DOI:https://doi.org/10.1137/0202019Google ScholarDigital Library
- Haim Kaplan, Shay Mozes, Yahav Nussbaum, and Micha Sharir. 2017. Submatrix maximum queries in Monge matrices and partial Monge matrices, and their applications. ACM Trans. Alg. 13, 2 (2017), 26. DOI:https://doi.org/10.1145/3039873Google Scholar
- Philip Klein. 2005. Multiple-source shortest paths in planar graphs. In Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA’05). SIAM, 146--155.Google ScholarDigital Library
- Philip N. Klein, Shay Mozes, and Christian Sommer. 2013. Structured recursive separator decompositions for planar graphs in linear time. In Proceedings of the 45th ACM Symposium on Theory of Computing (STOC’13). ACM, 505--514. DOI:https://doi.org/10.1145/2488608.2488672Google ScholarDigital Library
- Harold Kuhn. 1956. Variants of the Hungarian method for assignment problems. Nav. Res. Log. 3, 4 (1956), 253--258.Google ScholarCross Ref
- Richard J. Lipton and Robert Endre Tarjan. 1980. Applications of a planar separator theorem. SIAM J. Comput. 9, 3 (1980), 615--627. DOI:https://doi.org/10.1137/0209046Google ScholarDigital Library
- Shay Mozes and Christian Wulff-Nilsen. 2010. Shortest paths in planar graphs with real lengths in O(n log2 n/(log log n)). In Proceedings of the European Symposium on Algorithms (ESA’10). Springer, Berlin, 206--217. DOI:https://doi.org/10.1007/978-3-642-15781-3_18Google Scholar
- Marcin Mucha and Piotr Sankowski. 2004. Maximum matchings via Gaussian elimination. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS’04). IEEE, 248--255. DOI:https://doi.org/10.1109/FOCS.2004.40Google ScholarDigital Library
- Marcin Mucha and Piotr Sankowski. 2006. Maximum matchings in planar graphs via Gaussian elimination. Algorithmica 45, 1 (2006), 3--20. DOI:https://doi.org/10.1007/s00453-005-1187-5Google ScholarDigital Library
- L. Ramshaw and R. E. Tarjan. 2012. A weight-scaling algorithm for min-cost imperfect matchings in bipartite graphs. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE, 581--590. DOI:https://doi.org/10.1109/FOCS.2012.9.Google Scholar
- Piotr Sankowski. 2009. Maximum weight bipartite matching in matrix multiplication time. Theor. Comput. Sci. 410 (2009), 4480--4488. Issue 44. DOI:https://doi.org/10.1016/j.tcs.2009.07.028Google ScholarDigital Library
- R. Sharathkumar. 2013. A sub-quadratic algorithm for bipartite matching of planar points with bounded integer coordinates. In Proceedings of the 29th Symposium on Computational Geometry (SOCG’13). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 9--16. DOI:https://doi.org/10.1145/2462356.2480283Google ScholarDigital Library
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- A Faster Algorithm for Minimum-cost Bipartite Perfect Matching in Planar Graphs
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