Combinatorial optimization on embedded curves

We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compute certain similarity measures for embedded curves on a surface. Finally, we present algorithms to solve computational problems for compactly presented curves.;We describe the first algorithms to compute the shortest representative of a Z2-homology class. Given a directed graph embedded on a surface of genus g with b boundary cycles, we can compute the shortest single cycle Z2-homologous to a given even subgraph in 2O(g+b) nlog n time. As a consequence we obtain an algorithm to compute the shortest directed non-separating cycle in 2 O(g)n log n time, which improves the previous best algorithm by a factor of O(sqrt n) if the genus is a constant. Further, we can compute the shortest even subgraph in a given Z2-homology class if the input graph is undirected in the same asymptotic running time. As a consequence, we obtain the first near linear time algorithm to compute minimum ( s, t)-cuts in surface embedded graphs of constant genus. We also prove that computing the shortest even subgraph in a Z2 -homology class is in general NP-hard, which explains the exponential dependence on g..;We also consider the corresponding optimization problem under Z-homology. Given an integer circulation phi in a directed graph embedded on a surface of genus g, we describe algorithms to compute the minimum cost circulation that is Z-homologous to phi in O(g8n log 2 n log2 C) time if the capacities are integers whose sum is C or in g O(g)n3/2 time for arbitrary capacities. In particular, our algorithm improves the best known algorithm for computing the maximum (s, t)-flow on surface embedded graph after 20 years. The previous best algorithm, except for planar graphs, follow from general maximum flow algorithms for sparse graphs.;Next, we consider two closely related similarity measures of curves on piecewise linear surfaces embedded in R3, called homotopy height and homotopic Frechet distance. These similarity measures capture the longest curve that appears and the longest length that any point travels in the best morph between two given curves, respectively. We describe the first polynomial-time O(log n)-approximation algorithms for both problems. Prior to our work no algorithms were known for the homotopy height problem. For the homotopic Frechet distance, algorithms were known only for curves on Euclidean plane with polygonal obstacles. Surprisingly, it is not even known if deciding if either the homotopy height or the homotopic Frechet distance is smaller that a given value is in NP.;Finally, we consider normal curves on abstract triangulated surfaces. A curve is normal if it intersects any triangle in a finite set of arcs, each crossing between two different edges of the triangle. Given a triangulated surface of complexity n and a curve that crosses the triangulation X times, we can build another cell decomposition of the input surface of complexity O(n), in O(min(X, n2 log X)) time, whose 1-skeleton contains the input curve. We emphasize the the cell decomposition algorithm takes polynomial time even if X is exponential. The main ingredient of our cell decomposing algorithm is a technique to trace a curve in a triangulated surface. We apply our abstract tracing strategy to solve well-known problems about normal curves including computing the number of components, computing the number of isotopy classes and computing the algebraic intersection number between two curves. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms.