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16
SCIP: solving constraint integer programs
, 2009
"... Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), wh ..."
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Cited by 121 (0 self)
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Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and noncommercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly nonlinear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current stateoftheart techniques for proving the validity of properties on circuits containing arithmetic.
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
"... We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. ..."
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Cited by 23 (8 self)
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We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
Zigzag Persistent Homology in Matrix Multiplication Time
 IN: PROCEEDINGS OF THE TWENTYSEVENTH ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY, 2011
, 2011
"... We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our al ..."
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Cited by 20 (0 self)
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We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our algorithm runs in O(M(n) + n 2 log 2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n 2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n 3) time in the worst case.
Optimal topological simplification of discrete functions on surfaces
 Discrete & Computational Geometry
"... We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homol ..."
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Cited by 19 (6 self)
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We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2δ, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed. 1
Discrete Morse Theory for Manifolds with Boundary
, 2012
"... We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also ..."
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Cited by 14 (8 self)
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We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3andforeachk ≥ 0, there is a PL dsphere on which any discrete Morse function has more than k critical (d − 1)cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial dmanifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d − 1)cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible dball is
TADD: A Computational Framework for Data Analysis using Discrete Morse Theory
, 2010
"... This paper presents a computational framework that allows for a robust extraction of the extremal structure of scalar and vector fields on 2D manifolds embedded in 3D. This structure consists of critical points, separatrices, and periodic orbits. The framework is based on Forman’s discrete Morse t ..."
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Cited by 7 (5 self)
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This paper presents a computational framework that allows for a robust extraction of the extremal structure of scalar and vector fields on 2D manifolds embedded in 3D. This structure consists of critical points, separatrices, and periodic orbits. The framework is based on Forman’s discrete Morse theory, which guarantees the topological consistency of the computed extremal structure. Using a graph theoretical formulation of this theory, we present an algorithmic pipeline that computes a hierarchy of extremal structures. This hierarchy is defined by an importance measure and enables the user to select an appropriate level of detail.
Efficient computation of a hierarchy of discrete 3D gradient vector fields
 IN PROC. TOPOINVIS
, 2011
"... This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorit ..."
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Cited by 6 (3 self)
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This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorithm is based on Forman’s discrete Morse theory, which guarantees topological consistency and algorithmic robustness. In contrast to previous work, our algorithm combines memory and runtime efficiency. It thereby lends itself to the analysis of large data sets. A discrete gradient vector field is also a compact representation of the underlying extremal structures – the critical points, separation lines and surfaces. Given a certain level of detail, an explicit geometric representation of these structures can be extracted using simple and fast graph algorithms.
Discrete Morse functions from Fourier transforms
, 2007
"... A discrete Morse function for a simplicial complex describes how to construct a homotopy equivalent CWcomplex with hopefully fewer cells. We associate a boolean function with the simplicial complex and construct a discrete Morse function using its Fourier transform. Methods from theoretical compute ..."
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Cited by 5 (0 self)
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A discrete Morse function for a simplicial complex describes how to construct a homotopy equivalent CWcomplex with hopefully fewer cells. We associate a boolean function with the simplicial complex and construct a discrete Morse function using its Fourier transform. Methods from theoretical computer science by O’Donnell, Saks, Schramm, and Servedio, together with experimental data on complexes from Hachimoro’s library, provide some evidence that the constructed discrete Morse functions are efficient. 1