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Recently, Khuller, Moss and Naor presented a greedy algorithm for the budgeted maximum coverage problem. In this note, we observe that this algorithm also approximates a special case of set-union knapsack problem within a constant factor. In the special case, an element is a member of less than a constant number of subsets. This guarantee naturally extends to densest k-subgraph problem on graphs of bounded degree.
Deformable surface models are often represented as triangular meshes in image segmentation applications. For a fast and easily regularized deformation onto the target object boundary, the vertices of the mesh are commonly moved along line segments (typically surface normals). However, in case of high mesh curvature, these lines may intersect with the target boundary at “non-corresponding” positions, or may not intersect at all. Consequently, certain deformations cannot be achieved. We propose omnidirectional displacements for deformable surfaces (ODDS) to overcome this limitation. ODDS allow each vertex to move not only along a line segment but within a surrounding sphere, and achieve globally optimal deformations subject to local regularization con-
straints. However, allowing a ball-shaped instead of a linear range of motion per vertex significantly increases runtime and memory. To alleviate this drawback, we propose a hybrid approach, fastODDS, with improved runtime and reduced memory requirements. Furthermore, fastODDS can also cope with simultaneous segmentation of multiple objects. We show the theoretical benefits of ODDS with experiments on synthetic data, and evaluate ODDS and fastODDS quantitatively on clinical image data of the mandible and the hip bones. There, we assess both the global segmentation accuracy as well as local accuracy in high curvature regions, such as the tip-shaped mandibular coronoid processes and the ridge-shaped acetabular rims of
the hip bones.
Genome-scale metabolic networks are useful tools for achieving a system-level understanding of metabolism. However, due to their large size, analysis of such networks may be difficult and algorithms can be very slow. Therefore, some authors have suggested to analyze subsystems instead of the original genome-scale models.
Flux coupling analysis (FCA) is a well-known method for detecting functionally related reactions in metabolic networks.
In this paper, we study how flux coupling relations may change if we analyze a subsystem instead of the original network. We show mathematically that a pair of fully, partially or directionally coupled reactions may be detected as uncoupled in certain subsystems. Interestingly, this behavior is the opposite of the flux coupling changes that may occur due to missing reactions, or equivalently, deletion of reactions.
Computational experiments suggest that the analysis of plastid (but not mitochondrial) subsystems may significantly influence the results of FCA. Therefore, the results of FCA for subsystems, especially plastid subsystems, should be interpreted with care.
We introduce an algorithm for
diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both space and diffusion direction.
The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models.
It works by embedding the measurement space into a space with defined metric and group operations, in this case the Lie group of three-dimensional Euclidean motion SE(3).
Subsequently, pairwise comparisons of the values of the diffusion
weighted signal are used for adaptation.
The position-orientation adaptive smoothing preserves the edges of the observed fine and anisotropic structures.
The POAS-algorithm is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and
variability of quantities derived from the data for specific models.
We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and
hence acquisition time while achieving similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.
In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies or deformation energies. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved
surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.
We present a unified computational framework for matching 3d geometric objects (points, lines, surfaces, volumes) of highly varying shape. Our approach is based on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) method acting on $m$-currents. After stating an optimization algorithm in the function space of admissible morph generating velocity fields, two innovative aspects in this framework are presented: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Secondly, we directly compute the temporal evolution of discrete $m$-current attributes. Several numerical experiments demonstrate the effectiveness of this approach.
We derive and study a dynamical model for suspensions of negatively buoyant particles on an incline. Our theoretical model includes the settling/sedimentation due to gravity as well as the resuspension of particles induced by shear-induced migration, leading to disaggregation of the dense sediment layer. Out of the three different regimes observed in the experiments, we focus on the so-called settled case, where the particles settle out of the flow, and two distinct fronts, liquid and particle, form. Using an approach relying on asymptotics, we systematically connect our dynamic model with the previously developed equilibrium theory for particle-laden flows. We show that the resulting transport equations for the liquid and the particles are of hyperbolic type, and study the dilute limit, for which we derive the analytic solution. We also carry out a systematic experimental study of the settled regime, focusing on the motion of the liquid and the particle fronts. Finally, we carry out numerical simulations of our transport equations. We show that the model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.
We investigate different dynamical regimes of neuronal network in the CA3 area of the hippocampus.
The proposed neuronal circuit includes two fast- and two slowly-spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: gamma-rhythm in which the fast neurons are uniformly spiking, theta-rhythm in which the individual spikes are separated by quiet epochs, and theta/gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic strengths on the synchronization in the network and demonstrate that
strong asymmetry reduces the variety of available dynamical states. The model network exhibits multistability; this results in occurrence of hysteresis in dependence on the conductances of individual connections. We show that switching between different rhythmic patterns in the network
depends on the degree of synchronization between the slow cells.
Traffic in communication networks fluctuates heavily over time. Thus, to avoid capacity bottlenecks,
operators highly overestimate the traffic volume during network planning. In this paper we consider
telecommunication network design under traffic uncertainty, adapting the robust optimization approach
of Bertsimas and Sim [21]. We present three different mathematical formulations for this problem, provide valid inequalities,
study the computational implications, and evaluate the realized robustness.
To enhance the performance of the mixed-integer programming solver we derive robust cutset inequalities generalizing their deterministic counterparts. Instead of a single cutset inequality for every
network cut, we derive multiple valid inequalities by exploiting the extra variables available in the robust formulations. We show that these inequalities define facets under certain conditions and that they
completely describe a projection of the robust cutset polyhedron if the cutset consists of a single edge.
For realistic networks and live traffic measurements we compare the formulations and report on the
speed up by the valid inequalities. We study the “price of robustness” and evaluate the approach by
analyzing the real network load. The results show that the robust optimization approach has the potential
to support network planners better than present methods.
In this paper, we deal with a hydraulic reservoir optimization problem with uncertainty on
inflows in a joint chance constrained programming setting. In particular, we will consider inflows with
a persistency effect, following a causal time series model, and examine the impact of the ”Gaussian”
assumption for such inflows. We present an iterative algorithm for solving similarly structured joint
chance constrained programming problems that requires a Slater point and the computation of gradients.
Several alternatives to the joint chance constraint problem are presented. In particular, we present an
individual chance constraint problem and a robust model. We illustrate the interest of joint chance
constrained programming by comparing results obtained on a realistic hydro-valley with those obtained
from the alternative models. Despite the fact that the alternative models often require less hypothesis
on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models,
such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore
not useful. We therefore conclude that Joint Chance Constrained programming appears as a technique
offering a good trade-off between cost and robustness and can be tractable for complex realistic models.
Analysis of the second phase of the GMRES convergence for a convection-diffusion model problem
(2012)
It is well konwn that GMRES applied to linear algebraic systems arising from a convection-diffusion model problem that has been discretized by the streamline upwind Petrov-Galerkin (SUPG) method, typically displays two distinct phases of convergence: a slow initial phase followed by a convergence acceleration in the second phase. This paper complements the known results on the length of the initial phase by analyzing how the acceleration in the second phase of convergence is related to the mesh Peclet number and the choice of the stabilization parameter in the SUPG discretization. The analysis is based on some new expressions and bounds for the GMRES residuals, which can be of general interest.
The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combined-cycle-based cogeneration plant. The optimization problem is formulated as a nonconvex mixed-integer nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The algorithm generates a convex relaxation of the MINLP and applies a Branch and Cut algorithm to the relaxation. Numerical results for different demands for electric power and process steam are discussed and a sensitivity analysis is performed.
We provide lower estimates for the norm of gradients of Gaussian
distribution functions and apply the results obtained to a special class of
probabilistically constrained optimization problems. In particular, it is shown
how the precision of computing gradients in such problems can be controlled
by the precision of function values for Gaussian distribution functions. Moreover,
a sensitivity result for optimal values with respect to perturbations of the
underlying random vector is derived. It is shown that the so-called maximal
increasing slope of the optimal value with respect to the Kolmogorov distance
between original and perturbed distribution can be estimated explicitly from
the input data of the problem.
Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB~2010 benchmark instances.
Energetic reasoning is one of the most powerful propagation algorithms in cumulative scheduling. In practice, however, it is not commonly used because it has a high running time and its success highly depends on the tightness of the variable bounds. In order to speed up energetic reasoning, we provide an easy-to-check necessary condition for energetic reasoning to detect infeasibilities. We present an implementation of energetic reasoning that employs this condition and that can be parametrically adjusted to handle the trade-off between solving time and propagation overhead. Computational results on instances from the PSPLIB are provided. These results show that using this condition decreases the running time by more than a half, although more search nodes need to be explored.
We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.
Large neighborhood search (LNS) heuristics are an important component of modern branch-and-cut algorithms for solving mixed-integer linear programs (MIPs). Most of these LNS heuristics use the LP relaxation as the basis for their search, which is a reasonable choice in case of MIPs. However, for more general problem classes, the LP relaxation alone may not contain enough information about the original problem to find feasible solutions with these heuristics, e.g., if the problem is nonlinear or not all constraints are present in the current relaxation. In this paper, we discuss a generic way to extend LNS heuristics that have been developed for MIP to constraint integer programming (CIP), which is a generalization of MIP in the direction of constraint programming (CP). We present computational results of LNS heuristics for three problem classes: mixed-integer quadratically constrained programs, nonlinear pseudo-Boolean optimization instances, and resource-constrained project scheduling problems. Therefore, we have implemented extended versions of the following LNS heuristics in the constraint integer programming framework SCIP: Local Branching, RINS, RENS, Crossover, and DINS. Our results indicate that a generic generalization of LNS heuristics to CIP considerably improves the success rate of these heuristics.
We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLIB we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.
Learning during search allows solvers for discrete optimization problems to remember parts of the search that they have already performed and avoid revisiting redundant parts. Learning approaches pioneered by the SAT and CP communities have been successfully incorporated into the SCIP constraint integer programming platform. In this paper we show that performing a heuristic constraint programming search during root node processing of a binary program can rapidly learn useful nogoods, bound changes, primal solutions, and branching statistics that improve the remaining IP search.