Homology flows, cohomology cuts
Proceedings of the forty-first annual ACM symposium on Theory of computing
Randomly removing g handles at once
Proceedings of the twenty-fifth annual symposium on Computational geometry
Minimum cuts and shortest homologous cycles
Proceedings of the twenty-fifth annual symposium on Computational geometry
Randomly removing g handles at once
Computational Geometry: Theory and Applications
Maximum flows and parametric shortest paths in planar graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Global minimum cuts in surface embedded graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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By restricting the input to a problem, it often becomes possible to design more accurate or more efficient algorithms to solve that problem. In this thesis we restrict our attention to planar graphs and achieve both these goals. Planar graphs exhibit many structural and combinatorial properties that enable the design of good algorithms. These properties include: corresponding to every planar graph there is a dual planar graph; the dual of the complement of the edges of a spanning tree form a spanning tree of the dual graph; a set of edges is a cycle if and only if the dual edges form a cut; cycles can be said to enclose edges, faces and vertices in the planar embedding; paths can be compared as to their relative embedding. We capitalize on these properties to design (a) faster algorithms for polynomial-time-solvable network flow problems and (b) algorithms with better approximation guarantees for NP-hard connectivity problems. We give a conceptually simple O(n log n)-time algorithm for finding the maximum st-flow in a directed planar graph, proving a theorem that was incorrectly claimed over a decade ago. We also show how to compute the minimum cut between all pairs of vertices on a common face of a planar graph in linear time. We give the first polynomial-time approximation schemes for the Steiner-tree and 2-edge-connected subgraph problems. Both schemes are NP-hard in planar graphs and admit no PTAS in general graphs. Our schemes run in O(n log n) time.