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122
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
The Union of Balls and its Dual Shape
, 1993
"... Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived f ..."
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Cited by 183 (12 self)
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Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived from this complex. The algorithms are most relevant in R’3 where unions of finitely many balls are commonly used as models of molecules.
An Incremental Algorithm for Betti Numbers of Simplicial Complexes
, 1993
"... A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!i ..."
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Cited by 111 (14 self)
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A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!ied to the family of ashapes of a finite point set in R³ ittakes time O(ncz(n)) to compute the betti numbers of all crshapes.
Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 63 (13 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
Algorithmic Geometry of Numbers
 Annual Review of Comp. Sci
, 1987
"... this article  Algorithmic Geometry of Numbers. The fundamental basis reduction algorithm of Lov'asz which first appeared in Lenstra, Lenstra, Lov'asz [46] was used in Lenstra's algorithm for integer programming and has since been applied in myriad contextsstarting with factorization ..."
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this article  Algorithmic Geometry of Numbers. The fundamental basis reduction algorithm of Lov'asz which first appeared in Lenstra, Lenstra, Lov'asz [46] was used in Lenstra's algorithm for integer programming and has since been applied in myriad contextsstarting with factorization of polynomials (A.K. Lenstra, [45]). Classical Geometry of Numbers has a special feature in that it studies the geometric properties of (convex) sets like volume, width etc. which come from the realm of continuous mathematics in relation to lattices which are discrete objects. This makes it ideal for applications to integer programming and other discrete optimization problems which seem inherently harder than their "continuous" counterparts like linear programming. 1
Computing Betti Numbers via Combinatorial Laplacians
 ALGORITHMICA
, 1998
"... We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, ..."
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We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, of eigenvalues which we have yet to understand fully. We numerically verify a conjecture of Björner, Lovász, Vrećica, and ˘Zivaljevic ́ on the chessboard complexes C.4; 6/, C.5; 7/, and C.5; 8/. Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C.4; 6 / and C.5; 8/, giving a completely rigorous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.
Worstcase complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix
 SIAM J. Comput
, 1989
"... Abstract. An O(s5M($2)) algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size (this is the Smith normal form of the matrix) is presented. Moreover, an O(s3M(s2)) algorithm for computing the Hermite normal form of an integer matrix of size ..."
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Abstract. An O(s5M($2)) algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size (this is the Smith normal form of the matrix) is presented. Moreover, an O(s3M(s2)) algorithm for computing the Hermite normal form of an integer matrix of size is given. The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499507] and Chou and Collins in [SIAM J. Comput., 11 (1982), pp. 687708]. Key words. Smith normal form, Hermite normal form, integer matrices, computational complexity