Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with @math coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in @math so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.
In this work, we use graph cuts and their duality extensively. The duality of cuts on a planar graph and separating cycles on the dual graph has long been utilized to efficiently compute maximal flows and minimal cuts on planar graphs, a topic for which Chambers et al. @cite_11 provide a comprehensive review. In their paper @cite_11 , Chambers et al. discover the duality between minimal cuts of a surface-embedded graph and minimal homologous cycles in a dual complex, and then devise @math algorithms for both problems assuming the genus of the surface to be fixed. Chen and Freedman @cite_17 proposed an algorithm which computes a minimal non-bounding @math -cycle given a @math -complex embedded in @math , utilizing a natural duality of @math -cycles in the complex and cuts in the dual graph. The minimal non-bounding cycle algorithm can be further extended to solve the localization problem and the minimal basis problem over dimension @math given a @math -complex embedded in @math .
@cite_17 @cite_11
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We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut. We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s,t)-cut in gO(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP -hard. We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took O(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with O(m Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS’06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed. In the work of the authors (2003), we showed that graph cuts can find hypersurfaces of globally minimal length (or area) under any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significantly more general class of continuous non-symmetric metrics. Using submodularity condition (Boros and Hammer, 2002 and Kolmogorov and Zabih, 2004), we obtain a tight characterization of graph-representable metrics. Such "submodular" metrics have an elegant geometric interpretation via hypersurface functionals combining length area and flux. Practically speaking, we attend 'geo-cuts' algorithm to a wider class of geometrically motivated hypersurface functionals and show how to globally optimize any combination of length area and flux of a given vector field. The concept of flux was recently introduced into computer vision by Vasilevskiy and Siddiqi (2002) but it was mainly studied within variational framework so far. We are first to show that flux can be integrated into graph cuts as well. Combining geometric concepts of flux and length area within the global optimization framework of graph cuts allows principled discrete segmentation models and advances the slate of the art for the graph cuts methods in vision. In particular we address the "shrinking" problem of graph cuts, improve segmentation of long thin objects, and introduce useful shape constraints.