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SIAM J. on Matrix Analysis and Applications

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1987

Volume 8, Issue 4, pp. 519-808


SS/TDMA Satellite Communications with $k$-Permutation Switching Modes

J. L. Lewandowski and C. L. Liu

SIAM. J. on Algebraic and Discrete Methods 8, pp. 519-534 (16 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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The Satellite-Switched Time-Division Multiple Access (SS/TDMA) scheme has been one of the most effective techniques designed to allocate the communication bandwidth provided by communication satellites. The scheduling problem for SS/TDMA corresponds to finding a positive linear combination of a predefined set of (0,1)-matrices which covers a given traffic matrix ${\bf T}$ such that the sum of the multiplying constants used in the linear combination is minimum. In this paper, an algorithm is given to solve the optimization problem using a result which is a generalization of a theorem by Birkhoff and von Neumann. The case of $k$-permutation matrices is first addressed. The result is then further extended to more general sets of (0, 1)-matrices.

A Dynamic Programming Approach to the Dominating Set Problem on $k$-Trees

D. G. Corneil and J. M. Keil

SIAM. J. on Algebraic and Discrete Methods 8, pp. 535-543 (9 pages)

Online Publication Date: July 17, 2006

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Dynamic programming has long been established as an important technique for demonstrating the existence of polynomial time algorithms for various discrete optimization problems. In this paper we extend the normal paradigm of dynamic programming to allow a polynomial number of optimal solutions to be computed for each subproblem. This technique yields a polynomial time algorithm for the dominating set problem on $k$-trees, where $k$ is fixed. It is also shown that the dominating set problem is NP-complete for $k$-trees where $k$ is arbitrary.

The Null Space Problem II. Algorithms

Thomas F. Coleman and Alex Pothen

SIAM. J. on Algebraic and Discrete Methods 8, pp. 544-563 (20 pages) | Cited 22 times

Online Publication Date: July 17, 2006

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The null space problem is that of finding a sparsest basis for the null space (null basis) of an underdetermined matrix. This problem was shown to be NP-hard in Coleman and Pothen (this Journal, 7 (1986), pp. 527–537). In this paper we develop heuristic algorithms to find sparse null bases. A basis is computed by columns, i.e., by finding a null vector linearly independent of those previously obtained. The algorithms to compute null vectors have two phases. In the first combinatorial phase, a minimal dependent set of columns is identified by finding a matching in the bipartite graph of the matrix. In the second numerical phase, nonzero coefficients in the null vector are computed from this dependent set.
We have designed two algorithms: the first computes a fundamental basis (one with an embedded identity matrix), and the other, a triangular basis (one with an upper triangular matrix). We describe implementations of our algorithms and provide computational results on several large sparse constraint matrices from linear programs. Both algorithms find null bases which are quite sparse, have low running times, and require small intermediate storage. The triangular algorithm finds sparser bases at the expense of greater running times. We believe that this algorithm is an attractive candidate for large sparse null basis computations.

Embeddings of Ultrametric Spaces in Finite Dimensional Structures

Michael Aschbacher, Pierre Baldi, Eric B. Baum, and Richard M. Wilson

SIAM. J. on Algebraic and Discrete Methods 8, pp. 564-577 (14 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Motivated by recent advances in theoretical physics and combinatorial optimization, we study the problem of embedding ultrametric spaces into finite dimensional structures: finite sets, Euclidean spaces$\mathbb{R}^n $, Euclidean sphere $S^n $, and $n$-dimensional hypercube with Hamming distance. We give conditions and constructions of embeddings and show a general upper bound of $n + 1$ on the cardinality of the ultrametric set. We also give an upper bound on the cardinality of quasi-ultrametric sets.

The Exponent Set of Primitive, Nearly Reducible Matrices

Jia-Yu Shao

SIAM. J. on Algebraic and Discrete Methods 8, pp. 578-584 (7 pages)

Online Publication Date: July 17, 2006

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In [1] and [2], R. A. Brualdi and J. A. Ross studied the exponent set of a particular class of primitive matrices—primitive, nearly reducible matrices. They obtained an upper bound on the exponent and constructed some matrices with small exponents. Ross [2] suggested considering the problem of determining the quantity $e(n)$—the least integer $e(n)\geqq 6$ such that no $n \times n$ primitive, nearly reducible matrix has exponent $e(n)$. In this paper we give a nontrivial lower bound $e ( n )\geqq ( n^2 - 2n + 10 ) / 9 + 1$ by showing that every integer $k$ with $6\leqq k\leqq ( n^2 - 2n + 10 ) / 9$ is an exponent of some $n \times n$ primitive, nearly reducible matrix. This also extends the result ([2, § 3]) that every integer $k$ with $6\leqq k\leqq n + 1$ is the exponent of some $n \times n$ primitive, nearly reducible matrix.

Substitutes and Complements in Constrained Linear Models

J. Scott Provan

SIAM. J. on Algebraic and Discrete Methods 8, pp. 585-603 (19 pages)

Online Publication Date: July 17, 2006

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We consider the problem of when two variables in a linear programming model can be considered to be substitutes (self-interfering) or complements (self-reinforcing). Several definitions proposed in the economic and mathematical literature are investigated in the context of linear programming models. The concept of determinacy is used to formalize and classify these definitions. Determinacy is studied for a class of network flow models, where graph-theoretic characterizations of substitutes and complements are given.

On the Covering Radius Problem for Codes I. Bounds on Normalized Covering Radius

Karen E. Kilby and N. J. A. Sloane

SIAM. J. on Algebraic and Discrete Methods 8, pp. 604-618 (15 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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In this two-part paper we introduce the notion of a stable code and give a new upper bound on the normalized covering radius of a code. The main results are that, for fixed $k$ and large $n$, the minimal covering radius $t[n,k]$ is realized by a normal code in which all but one of the columns have multiplicity 1; hence $t[n + 2,k] = t[n,k] + 1$ for sufficiently large $n$. We also show that codes with $n\leqq 14, k\leqq 5$ or $d_{\min } \leqq 5$ are normal, and we determine the covering radius of all proper codes of dimension $k\leqq 5$. Examples of abnormal nonlinear codes are given. In Part I we investigate the general theory of normalized covering radius, while in Part II [this Journal, 8 (1987), pp. 619–627] we study codes of dimension $k\leqq 5$, and normal and abnormal codes.

On the Covering Radius Problem for Codes II. Codes of Low Dimension; Normal and Abnormal Codes

Karen E. Kilby and N. J. A. Sloane

SIAM. J. on Algebraic and Discrete Methods 8, pp. 619-627 (9 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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In this two-part paper we introduce the notion of a stable code and give a new upper bound on the normalized covering radius of a code. The main results are that, for fixed $k$ and large $n$, the minimal covering radius $t[n,k]$ is realized by a normal code in which all but one of the columns have multiplicity 1; hence$t[n + 2,k] = t[n,k] + 1$ for sufficiently large $n$. We also show that codes with $n\leqq 14, k\leqq 5$ or $d_{\min } \leqq 5$ are normal, and we determine the covering radius of all proper codes of dimension $k\leqq 5$. Examples of abnormal nonlinear codes are given. In Part I [this Journal, 8 (1987), pp. 604–618] we investigated the general theory of normalized covering radius, while in Part II we study codes of dimension $k\leqq 5$, and normal and abnormal codes.

Latent Subsets for Dual Intersecting Systems

P. C. Fishburn

SIAM. J. on Algebraic and Discrete Methods 8, pp. 628-634 (7 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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A collection $F$ of subsets of $\{ 1,2, \cdots ,n\} $ is a dual intersecting system if no two sets in $F$ have an empty intersection or an exhaustive union. Let $f_j $ be the number of sets in $F$ that contain point $j$, and let $f_j^L $ be the number of sets not in $F$ but included in some set in $F$ that contain $j$.
We conjecture that if $F$ is a dual intersecting system, then $f_j^L \geqq f_j $ for some $j$ in $\{ 1, \cdots ,n\} $. This is shown to be true if either min $f_j \leqq n$ or min $f_j < 8$.

Extremal Length and Width of Blocking Polyhedra, Kirchhoff Spaces and Multiport Networks

Seth Chaiken

SIAM. J. on Algebraic and Discrete Methods 8, pp. 635-645 (11 pages)

Online Publication Date: July 17, 2006

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Various facts about the extremal length (EL) and extremal width (EW) of a one-port network on a Kirchhoff space due to Anderson, Duffin and Trapp and their relation to blocking pairs of polyhedra are unified and extended to the multiport case. The definitions of EL and EW are extended to all pairs of blocking polyhedra $(G,H)$ on coordinates $E$ given a symmetric positive definite matrix $R$. It follows that $\text{EW}^{ -1} = \min \{ x^t Rx | x \in G \} , \text{EL}^{ -1} = \min \{ z^t R^{-1} z | z \in H \} $ and $\text{EL} \cdot \text{EW} = 1$. A Kirchhoff space on coordinates $(E,P)$ where $P$ is called the set of ports is a subspace that represents a matroid on $E \cup P$ in which $P$ is independent and co-independent. Given any nonzero vector $\omega $ on port coordinates $P$, we extend Fulkerson’s construction of a blocking pair from orthogonal subspaces with one distinguished coordinate to Kirchhoff spaces which model multiport networks. For $\omega $ and positive definite $R$ a pair of minimization problems with reciprocal values are derived from Kirchhoff spaces. When $R$ is diagonal these problems coincide with the $\text{EW}^{ - 1} $ and $\text{EL}^{ - 1} $ problems for the blocking pair from Kirchhoff spaces. In the case of a multiport resistor network, EW is the power dissipated when the voltage vector $\omega $ is applied to the ports.

A New Heuristic for Minimum Weight Triangulation

Andrzej Lingas

SIAM. J. on Algebraic and Discrete Methods 8, pp. 646-658 (13 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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A new heuristic for minimum weight triangulation of planar point sets is proposed. First, a polygon whose vertices are all points from the input set is constructed. Next, a minimum weight triangulation of the polygon is found by dynamic programming. The union of the polygon triangulation with the polygon yields a triangulation of the input $n$-point set. A nontrivial upper bound on the worst-case performance of the heuristic in terms of $n$ and another parameter is derived. Under the assumption of uniform point distribution it is observed that the heuristic yields a solution within the factor of $O(\log n)$ from the optimum almost certainly, and the expected length of the resulting triangulation is of the same order as that of a minimum length triangulation. The heuristic runs in time $O(n^3 )$ .

On Minimum Critically $n$-Edge-Connected Graphs

Margaret B. Cozzens and Shu-Shih Y. Wu

SIAM. J. on Algebraic and Discrete Methods 8, pp. 659-669 (11 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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Let $n$ be an integer with $n\geqq 2$. A graph $G$ is called critically $n$-edge-connected if the edge-connectivity $\lambda (G) = n$ and for any vertex $v$ of $G$, $\lambda (G - \upsilon ) = n - 1$. The sizes of critically $n$-edge-connected graphs are important and interesting in applications in communication networks. The maximum graphs with this property have been characterized [2]. In this paper, we first discuss some properties of minimum graphs, then show that the problem of finding a minimum critically $n$-edge-connected spanning subgraph of a given graph $G$ is NP-complete.

Four Variational Formulations of the Contraharmonic Mean of Operators

W. L. Green and T. D. Morley

SIAM. J. on Algebraic and Discrete Methods 8, pp. 670-673 (4 pages)

Online Publication Date: July 17, 2006

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The authors establish four variational expressions, each related to the parallel sum, for the contraharmonic mean of two positive operators or matrices.

The Contraharmonic Mean of HSD Matrices

William N. Anderson, Jr., Michael E. Mays, Thomas D. Morley, and George E. Trapp

SIAM. J. on Algebraic and Discrete Methods 8, pp. 674-682 (9 pages)

Online Publication Date: July 17, 2006

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For positive scalars $a$ and $b$ the contraharmonic mean of $a$ and $b$, $C(a,b)$, is defined by \[ C(a,b) = (a^2 + b^2 )/(a + b). \] In this paper we consider a natural matrix generalization of the contraharmonic mean, fit this into the matrix analogue of some of the classical scalar inequalities for means, develop computational procedures which let us generate the matrix analogues of an infinite family of scalar means, and study fixed point problems. Finally, we mention a relationship between least squares problems and the contraharmonic mean.

Fast Parallel Computation of Hermite and Smith Forms of Polynomial Matrices

Erich Kaltofen, M. S. Krishnamoorthy, and B. David Saunders

SIAM. J. on Algebraic and Discrete Methods 8, pp. 683-690 (8 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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Boolean circuits of polynomial size and polylogarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficientsequential algorithms. Furthermore, we give a polynomial-time deterministic sequential algorithm for the Smith normal form over the rationals. The Smith normal form algorithms are applied to the rational canonical form of matrices over finite fields and the field of rational numbers.

The Case of Equality in Hopf’s Inequality

Lorenzo O. Hilliard

SIAM. J. on Algebraic and Discrete Methods 8, pp. 691-709 (19 pages)

Online Publication Date: July 17, 2006

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Hopf’s inequality states that the subdominant eigenvalues $\lambda $ of a positive $n$-square matrix $A$ satisfy \[ | \lambda |\leqq \frac{M - m}{M + m} \lambda _p \]where $\lambda _p $ is the Perron eigenvalue of $A$ and $M$, $m$ are, respectively, the maximum and minimum entries of $A$. A complete analysis of the case of equality in Hopf’s inequality is given. If $A$ has an eigenvalue $\lambda $ which satisfies the case of equality, it is shown that $\lambda $ is real and the structure of the matrix $A$ is determined.

The General Minimum Fill-In Problem

H. Wendel

SIAM. J. on Algebraic and Discrete Methods 8, pp. 710-745 (36 pages)

Online Publication Date: July 17, 2006

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We consider the well-known graph-theoretical elimination process which is related to Gaussian elimination on a sparse, positive definite system of linear equations. The general minimum fill-in problem isconcerned in ranking (elimination) orderings by so-called criterion functions and is interested in those orderings which minimize (which are optimal with respect to) any (fixed) criterion function or, more generally, whichminimize even the whole class of such functions. A valuable tool for attacking this problem is the Initial Theorem due to Bertele and Brioschi (J. Math. Anal. Appl., 35 (1971), pp. 48–57). An Isomorphic Theorem can be proved guaranteeing a particular invariance property which is of great importance for the application of the Initial Theorem. In addition we consider the so-called separation approach which—roughly speaking—splits a given graph $G$ into two partial graphs $G_1 $ and $G_2 $ so that optimal orderings of $G_1 $ and $G_2 $ together form an optimal ordering of $G$. We are able to give conditions on a separating set of vertices sufficient for this procedure. Furthermore, a special class of graphs is introduced which arise in the field of load-flow calculation. The Initial Theorem is generalized to that class of graphs.

Mixing Rates for a Random Walk on the Cube

Peter Matthews

SIAM. J. on Algebraic and Discrete Methods 8, pp. 746-752 (7 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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For a simple random walk on the cube a coupling and a strong uniform time are given. The coupling gives an upper bound on the variation distance between the distribution after $k$ steps and the uniform distribution that is almost the best possible. The strong uniform time is used to calculate the variation distance and the separation. The coupling and strong uniform time are intimately related to a hitting time for the Ehrenfest chain and the time taken by a random graph to become connected, respectively.

An Algebraic Construction of Sonar Sequences Using $M$-Sequences

Richard A. Games

SIAM. J. on Algebraic and Discrete Methods 8, pp. 753-761 (9 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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An algebraic construction of sonar sequences that is based on the properties of $q$-ary $M$ -sequences for $q$ a prime power is presented. Sonar sequences give two-dimensional synchronization patterns that have two-dimensional spatial aperiodic autocorrelation functions with minimum out-of-phase values. The best sonarsequences with length $q^m \leqq 128$ that are obtained from the construction are tabulated. Based on a comparison with the limited number of known optimal values, the construction performs quite well, producing optimal sonar sequences in the majority of applicable cases.

Some Completeness Results on Decision Trees and Group Testing

Ding-Zhu Du and Ker-I Ko

SIAM. J. on Algebraic and Discrete Methods 8, pp. 762-777 (16 pages)

Online Publication Date: July 17, 2006

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The computational complexity of the group testing problem is investigated under the minimax measure and the decision tree model. We consider the generalizations of the group testing problem in which partial information about the decision tree of the problem is given. Using this approach, we demonstrate theNP-hardness of several decision problems related to various models of the group testing problem. For example, we show that, for several models of group testing, the problem of recognizing a set of queries that uniquely determines each object is co-NP-complete.

A Recursive Hadamard Transform Optimal Soft Decision Decoding Algorithm

Yair Be’ery and Jakov Snyders

SIAM. J. on Algebraic and Discrete Methods 8, pp. 778-789 (12 pages)

Online Publication Date: July 17, 2006

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A recursive soft decision maximum likelihood Hadamard transform decoding rule for binary code is derived. This algorithm, with computational complexity that varies inversely with the code rate for a fixed code length, is efficiently applicable for decoding convolutional codes and high rate block codes. An even more significant reduction in decoder complexity is obtained when the algorithm is applied for decoding product codes and concatenated codes. This algorithm achieves the computational efficiency of the Viterbi algorithm. In addition, its structural regularity simplifies the VLSI implementation of decoders.

On One-Sided Jacobi Methods for Parallel Computation

P. J. Eberlein

SIAM. J. on Algebraic and Discrete Methods 8, pp. 790-796 (7 pages)

Online Publication Date: July 17, 2006

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Convergence proofs are given for one-sided Jacobi/Hestenes methods for the singular value problem. The limiting form of the matrix iterates for the Hestenes method with optimization when the original matrix is normal is derived; this limiting matrix is block diagonal, where the blocks are multiples of unitary matrices. A variation in the algorithm to guarantee convergence to a diagonal matrix for the symmetric eigenvalue problem is shown. Implementation techniques for parallel computation, in particular, on the hypercube are indicated.

An Efficient Factorization for the Group Inverse

Bernard F. Lamond

SIAM. J. on Algebraic and Discrete Methods 8, pp. 797-808 (12 pages)

Online Publication Date: July 17, 2006

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An efficient algorithm is introduced for computing the group inverse of a square, singular matrix, in factorized form. The algorithm is based on the QR factorization with column pivoting and uses a technique of inversion by partitioning. The factorization is used to compute the group inverse solution of a singular system of equations. When only the solution vector is wanted, the group inverse does not need to be computed explicitly.
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