@inproceedings{2807,
  abstract     = {We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex. },
  author       = {Čadek, Martin and Krcál, Marek and Matoušek, Jiří and Vokřínek, Lukáš and Wagner, Uli},
  booktitle    = {45th Annual ACM Symposium on theory of computing},
  location     = {Palo Alto, CA, United States},
  pages        = {595 -- 604},
  publisher    = {ACM},
  title        = {{Extending continuous maps: Polynomiality and undecidability}},
  doi          = {10.1145/2488608.2488683},
  year         = {2013},
}

@inproceedings{2812,
  abstract     = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H* (X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.},
  author       = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André},
  booktitle    = {Proceedings of the 29th annual symposium on Computational Geometry},
  location     = {Rio de Janeiro, Brazil},
  pages        = {117 -- 125},
  publisher    = {ACM},
  title        = {{Homological reconstruction and simplification in R3}},
  doi          = {10.1145/2462356.2462373},
  year         = {2013},
}

@article{2815,
  abstract     = {The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in ℝ1 and are generally not well studied in higher dimensions. We study a configuration of n+1 Gaussian kernels for which there are exactly n+2 modes. We show that all modes lie on a finite set of lines, which we call axes, and study the restriction of the Gaussian mixture to these axes in order to discover that there are an exponential number of critical points in this configuration. Although the existence of ghost modes remained unknown due to the difficulty of finding examples in ℝ2, we show that the resilience of ghost modes grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.},
  author       = {Edelsbrunner, Herbert and Fasy, Brittany Terese and Rote, Günter},
  issn         = {1432-0444},
  journal      = {Discrete & Computational Geometry},
  number       = {4},
  pages        = {797 -- 822},
  publisher    = {Springer},
  title        = {{Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions}},
  doi          = {10.1007/s00454-013-9517-x},
  volume       = {49},
  year         = {2013},
}

@article{2822,
  abstract     = {Identification of genes that control root system architecture in crop plants requires innovations that enable high-throughput and accurate measurements of root system architecture through time. We demonstrate the ability of a semiautomated 3D in vivo imaging and digital phenotyping pipeline to interrogate the quantitative genetic basis of root system growth in a rice biparental mapping population, Bala x Azucena. We phenotyped >1,400 3D root models and >57,000 2D images for a suite of 25 traits that quantified the distribution, shape, extent of exploration, and the intrinsic size of root networks at days 12, 14, and 16 of growth in a gellan gum medium. From these data we identified 89 quantitative trait loci, some of which correspond to those found previously in soil-grown plants, and provide evidence for genetic tradeoffs in root growth allocations, such as between the extent and thoroughness of exploration. We also developed a multivariate method for generating and mapping central root architecture phenotypes and used it to identify five major quantitative trait loci (r2 = 24-37%), two of which were not identified by our univariate analysis. Our imaging and analytical platform provides a means to identify genes with high potential for improving root traits and agronomic qualities of crops.},
  author       = {Topp, Christopher and Iyer Pascuzzi, Anjali and Anderson, Jill and Lee, Cheng and Zurek, Paul and Symonova, Olga and Zheng, Ying and Bucksch, Alexander and Mileyko, Yuriy and Galkovskyi, Taras and Moore, Brad and Harer, John and Edelsbrunner, Herbert and Mitchell Olds, Thomas and Weitz, Joshua and Benfey, Philip},
  journal      = {PNAS},
  number       = {18},
  pages        = {E1695 -- E1704},
  publisher    = {National Academy of Sciences},
  title        = {{3D phenotyping and quantitative trait locus mapping identify core regions of the rice genome controlling root architecture}},
  doi          = {10.1073/pnas.1304354110},
  volume       = {110},
  year         = {2013},
}

@inproceedings{2843,
  abstract     = {Mathematical objects can be measured unambiguously, but not so objects from our physical world. Even the total length of tubelike shapes has its difficulties. We introduce a combination of geometric, probabilistic, and topological methods to design a stable length estimate for tube-like shapes; that is: one that is insensitive to small shape changes.},
  author       = {Edelsbrunner, Herbert and Pausinger, Florian},
  booktitle    = {17th IAPR International Conference on Discrete Geometry for Computer Imagery},
  location     = {Seville, Spain},
  pages        = {XV -- XIX},
  publisher    = {Springer},
  title        = {{Stable length estimates of tube-like shapes}},
  doi          = {10.1007/978-3-642-37067-0},
  volume       = {7749},
  year         = {2013},
}

@article{2887,
  abstract     = {Root system growth and development is highly plastic and is influenced by the surrounding environment. Roots frequently grow in heterogeneous environments that include interactions from neighboring plants and physical impediments in the rhizosphere. To investigate how planting density and physical objects affect root system growth, we grew rice in a transparent gel system in close proximity with another plant or a physical object. Root systems were imaged and reconstructed in three dimensions. Root-root interaction strength was calculated using quantitative metrics that characterize the extent towhich the reconstructed root systems overlap each other. Surprisingly, we found the overlap of root systems of the same genotype was significantly higher than that of root systems of different genotypes. Root systems of the same genotype tended to grow toward each other but those of different genotypes appeared to avoid each other. Shoot separation experiments excluded the possibility of aerial interactions, suggesting root communication. Staggered plantings indicated that interactions likely occur at root tips in close proximity. Recognition of obstacles also occurred through root tips, but through physical contact in a size-dependent manner. These results indicate that root systems use two different forms of communication to recognize objects and alter root architecture: root-root recognition, possibly mediated through root exudates, and root-object recognition mediated by physical contact at the root tips. This finding suggests that root tips act as local sensors that integrate rhizosphere information into global root architectural changes.},
  author       = {Fang, Suqin and Clark, Randy and Zheng, Ying and Iyer Pascuzzi, Anjali and Weitz, Joshua and Kochian, Leon and Edelsbrunner, Herbert and Liao, Hong and Benfey, Philip},
  journal      = {PNAS},
  number       = {7},
  pages        = {2670 -- 2675},
  publisher    = {National Academy of Sciences},
  title        = {{Genotypic recognition and spatial responses by rice roots}},
  doi          = {10.1073/pnas.1222821110},
  volume       = {110},
  year         = {2013},
}

@inproceedings{2906,
  abstract     = {Motivated by an application in cell biology, we describe an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. Our algorithm is implemented
using CGAL, following the exact geometric computation paradigm. We report on several
techniques to accelerate the computation that turn our implementation applicable to the underlying biological
problem.},
  author       = {Kerber, Michael and Edelsbrunner, Herbert},
  booktitle    = {2013 Proceedings of the 15th Workshop on Algorithm Engineering and Experiments},
  location     = {New Orleans, LA, United States},
  pages        = {70 -- 77},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{3D kinetic alpha complexes and their implementation}},
  doi          = {10.1137/1.9781611972931.6},
  year         = {2013},
}

@article{2939,
  abstract     = {In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ > 0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0, 1), the running time is O (C (1 - δ) Γ R d (n) log n), where C (1 - δ) Γ is the number of homology classes with persistence at least (1 - δ) Γ, n is the total number of simplices in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} > 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n).},
  author       = {Chen, Chao and Kerber, Michael},
  journal      = {Computational Geometry: Theory and Applications},
  number       = {4},
  pages        = {435 -- 447},
  publisher    = {Elsevier},
  title        = {{An output sensitive algorithm for persistent homology}},
  doi          = {10.1016/j.comgeo.2012.02.010},
  volume       = {46},
  year         = {2013},
}

@inproceedings{2901,
  abstract     = { We introduce the M-modes problem for graphical models: predicting the M label configurations of highest probability that are at the same time local maxima of the probability landscape. M-modes have multiple possible applications: because they are intrinsically diverse, they provide a principled alternative to non-maximum suppression techniques for structured prediction, they can act as codebook vectors for quantizing the configuration space, or they can form component centers for mixture model approximation. We present two algorithms for solving the M-modes problem. The first algorithm solves the problem in polynomial time when the underlying graphical model is a simple chain. The second algorithm solves the problem for junction chains. In synthetic and real dataset, we demonstrate how M-modes can improve the performance of prediction. We also use the generated modes as a tool to understand the topography of the probability distribution of configurations, for example with relation to the training set size and amount of noise in the data. },
  author       = {Chen, Chao and Kolmogorov, Vladimir and Yan, Zhu and Metaxas, Dimitris and Lampert, Christoph},
  booktitle    = {Proceedings of the 16th International Conference on Artificial Intelligence and Statistics},
  location     = {Scottsdale, AZ, United States},
  pages        = {161 -- 169},
  publisher    = {Journal of Machine Learning Research},
  title        = {{Computing the M most probable modes of a graphical model}},
  volume       = {31},
  year         = {2013},
}

@article{2859,
  abstract     = {Given a continuous function f:X-R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization.},
  author       = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
  journal      = {Homology, Homotopy and Applications},
  number       = {1},
  pages        = {51 -- 72},
  publisher    = {International Press of Boston},
  title        = {{Homology and robustness of level and interlevel sets}},
  doi          = {10.4310/HHA.2013.v15.n1.a3},
  volume       = {15},
  year         = {2013},
}

@article{2849,
  author       = {Edelsbrunner, Herbert and Strelkova, Nataliya},
  journal      = {Russian Mathematical Surveys},
  number       = {6},
  pages        = {1167 -- 1168},
  publisher    = {IOP Publishing},
  title        = {{On the configuration space of Steiner minimal trees}},
  doi          = {10.1070/RM2012v067n06ABEH004820},
  volume       = {67},
  year         = {2012},
}

@article{2902,
  abstract     = {We present an algorithm for simplifying linear cartographic objects and results obtained with a computer program implementing this algorithm. },
  author       = {Edelsbrunner, Herbert and Musin, Oleg and Ukhalov, Alexey and Yakimova, Olga and Alexeev, Vladislav and Bogaevskaya, Victoriya and Gorohov, Andrey and Preobrazhenskaya, Margarita},
  journal      = {Modeling and Analysis of Information Systems},
  number       = {6},
  pages        = {152 -- 160},
  publisher    = {Russian Academy of Sciences},
  title        = {{Fractal and computational geometry for generalizing cartographic objects}},
  volume       = {19},
  year         = {2012},
}

@inproceedings{2903,
  abstract     = {In order to enjoy a digital version of the Jordan Curve Theorem, it is common to use the closed topology for the foreground and the open topology for the background of a 2-dimensional binary image. In this paper, we introduce a single topology that enjoys this theorem for all thresholds decomposing a real-valued image into foreground and background. This topology is easy to construct and it generalizes to n-dimensional images.},
  author       = {Edelsbrunner, Herbert and Symonova, Olga},
  location     = {New Brunswick, NJ, USA },
  pages        = {41 -- 48},
  publisher    = {IEEE},
  title        = {{The adaptive topology of a digital image}},
  doi          = {10.1109/ISVD.2012.11},
  year         = {2012},
}

@article{2904,
  abstract     = {Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P(i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b - 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.},
  author       = {Pausinger, Florian},
  issn         = {2118-8572},
  journal      = {Journal de Theorie des Nombres des Bordeaux},
  number       = {3},
  pages        = {729 -- 749},
  publisher    = {Université de Bordeaux},
  title        = {{Weak multipliers for generalized van der Corput sequences}},
  doi          = {10.5802/jtnb.819},
  volume       = {24},
  year         = {2012},
}

@article{2912,
  author       = {Edelsbrunner, Herbert and Strelkova, Nataliya},
  journal      = {Russian Mathematical Surveys},
  number       = {6},
  pages        = {1167–1168},
  publisher    = {Russian Academy of Sciences},
  title        = {{On the configuration space for the shortest networks}},
  doi          = {10.4213/rm9503},
  volume       = {67},
  year         = {2012},
}

@article{2941,
  author       = {Dolbilin, Nikolai and Edelsbrunner, Herbert and Musin, Oleg},
  journal      = {Russian Mathematical Surveys},
  number       = {4},
  pages        = {781 -- 783},
  publisher    = {IOP Publishing},
  title        = {{On the optimality of functionals over triangulations of Delaunay sets}},
  doi          = {10.1070/RM2012v067n04ABEH004807},
  volume       = {67},
  year         = {2012},
}

@inproceedings{2971,
  abstract     = {We study the task of interactive semantic labeling of a segmentation hierarchy. To this end we propose a framework interleaving two components: an automatic labeling step, based on a Conditional Random Field whose dependencies are defined by the inclusion tree of the segmentation hierarchy, and an interaction step that integrates incremental input from a human user. Evaluated on two distinct datasets, the proposed interactive approach efficiently integrates human interventions and illustrates the advantages of structured prediction in an interactive framework. },
  author       = {Zankl, Georg and Haxhimusa, Yll and Ion, Adrian},
  booktitle    = {34th DAGM and 36th OAGM Symposium},
  issn         = {1611-3349},
  location     = {Graz, Austria},
  pages        = {11 -- 20},
  publisher    = {Springer},
  title        = {{Interactive labeling of image segmentation hierarchies}},
  doi          = {10.1007/978-3-642-32717-9_2},
  volume       = {7476},
  year         = {2012},
}

@article{6588,
  abstract     = {First we note that the best polynomial approximation to vertical bar x vertical bar on the set, which consists of an interval on the positive half-axis and a point on the negative half-axis, can be given by means of the classical Chebyshev polynomials. Then we explore the cases when a solution of the related problem on two intervals can be given in elementary functions.},
  author       = {Pausinger, Florian},
  issn         = {1812-9471},
  journal      = {Journal of Mathematical Physics, Analysis, Geometry},
  number       = {1},
  pages        = {63--78},
  publisher    = {B. Verkin Institute for Low Temperature Physics and Engineering},
  title        = {{Elementary solutions of the Bernstein problem on two intervals}},
  volume       = {8},
  year         = {2012},
}

@article{3115,
  abstract     = {We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(nlogn)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.},
  author       = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza},
  journal      = {Discrete & Computational Geometry},
  number       = {4},
  pages        = {964 -- 989},
  publisher    = {Springer},
  title        = {{Deconstructing approximate offsets}},
  doi          = {10.1007/s00454-012-9441-5},
  volume       = {48},
  year         = {2012},
}

@article{3120,
  abstract     = {We introduce a strategy based on Kustin-Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9 × 16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology. © 2012 Copyright Foundation Compositio Mathematica.},
  author       = {Brown, Gavin and Kerber, Michael and Reid, Miles},
  journal      = {Compositio Mathematica},
  number       = {4},
  pages        = {1171 -- 1194},
  publisher    = {Cambridge University Press},
  title        = {{Fano 3 folds in codimension 4 Tom and Jerry Part I}},
  doi          = {10.1112/S0010437X11007226},
  volume       = {148},
  year         = {2012},
}