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Divisionfree algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches
 COMPUTATIONAL DISCRETE MATHEMATICS
, 2001
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Computing invariants of simplicial manifolds
, 2004
"... Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic ..."
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Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book “Triangulated Manifolds with Few Vertices ” by Frank H. Lutz. The purpose of this chapter is to survey what is known about algorithms for the computation of algebraic invariants of topological spaces. Primarily, we use finite simplicial complexes as our model of topological spaces; for a discussion of different views see Section 4. On the way we give explicit definitions or constructions of all invariants presented. Note that we did not try to phrase all the results in their greatest generality. Similarly, we focus on invariants for which actual implementations exist. The reader is referred to Bredon’s monograph [2] for the wider perspective. For a related survey see Vegter [44]. 1. Homology
On the degree of mixed polynomial matrices
 SIAM Journal on Matrix Analysis and Applications
, 1999
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Deterministic Polynomial Time Algorithms for Matrix Completion Problems
, 2009
"... We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundament ..."
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We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundamental problems in computational complexity with numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [HKM05, HKY06]. We design efficient deterministic algorithms for common generalizations of the results of Lovász and Geelen on this problem by allowing linear functions in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. We present also a deterministic polynomial time algorithm for finding the minimal number of generators of a given module structure given by matrices. We establish further several hardness results related to matrix algebras and modules. As a result we connect the classical problem of polynomial identity testing with checking surjectivity (or injectivity) between two given modules. One of the elements of our algorithm is a construction of a greedy algorithm for finding a maximum rank element in the more general setting of the problem. The proof methods used in this paper could be also of independent interest.
A computational basis for higherdimensional computational geometry and applications
 COMPUTATIONAL GEOMETRY
, 1998
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Combinatorial Optimization: A Survey
, 1993
"... This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which the ..."
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This paper is a chapter of the forthcoming Handbook of Combinatorics, to be published by NorthHolland. It surveys the basic techniques and methods in combinatorial optimization. We organize our material according to the fundamental algorithmic techniques and illustrate them on problems to which these methods have been applied successfully. Special attention is given to approximation algorithms and fast (primal and dual) heuristics.
Parallel Complexity of Computations with General and Toeplitzlike Matrices Filled with Integers and Extensions
, 1999
"... 1 Computations with Toeplitz and Toeplitzlike matrices are fundamental for many areas of algebraic and numerical computing. The list of computational problems reducible to Toeplitz and Toeplitzlike computations includes, in particular, the evaluation of the gcd, the lcm, and the resultant of two p ..."
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1 Computations with Toeplitz and Toeplitzlike matrices are fundamental for many areas of algebraic and numerical computing. The list of computational problems reducible to Toeplitz and Toeplitzlike computations includes, in particular, the evaluation of the gcd, the lcm, and the resultant of two polynomials, computing Pad'e approximation and the BerlekampMassey recurrence coefficients, as well as numerous problems reducible to these. Transition to Toeplitz and Toeplitzlike computations is currently the basis for the design of the fastest known parallel (RNC) algorithms for these computational problems. Our main result is in contructing nearly optimal randomized parallel algorithms for Toeplitz and Toeplitzlike computations and, consequently, for numerous related computational problems (including the computational problems listed above), where all the input values are integers and all the output values are computed exactly. This includes randomized parallel algorithms for computing...
Totally Tight ChvátalGomory Cuts
, 2002
"... Let P := fx 2 IR : Ax bg be a polyhedron and PI its integral hull. A ChvatalGomory (CG) cut is a valid inequality for PI of the form bc, with 2 IR + , b = 2 Z. We give a polynomialtime algorithm which, given some x 2 P , detects whether a totally tight CG cut exists, i.e., whether there is a CG ..."
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Let P := fx 2 IR : Ax bg be a polyhedron and PI its integral hull. A ChvatalGomory (CG) cut is a valid inequality for PI of the form bc, with 2 IR + , b = 2 Z. We give a polynomialtime algorithm which, given some x 2 P , detects whether a totally tight CG cut exists, i.e., whether there is a CG cut such that A)x = b. Such a CG cut is violated by as much as possible under the assumption that x 2 P .