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SIAM J. on Matrix Analysis and Applications

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1982

Volume 3, Issue 4, pp. 411-605


Rearrangeable Networks with Limited Depth

Nicholas Pippenger and Andrew C.-C. Yao

SIAM. J. on Algebraic and Discrete Methods 3, pp. 411-417 (7 pages) | Cited 10 times

Online Publication Date: July 17, 2006

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Rearrangeable networks are switching systems capable of establishing simultaneous independent communication paths in accordance with any one-to-one correspondence between their $n$ inputs and $n$ outputs. Classical results show that $\Omega ( n \log n )$ switches are necessary and that $O ( n \log n )$ switches are sufficient for such networks. We are interested in the minimum possible number of switches in rearrangeable networks in which the depth (the length of the longest path from an input to an output) is at most $k$, where $k$ is fixed as $n$ increases. We show that $\Omega ( n^{1 + 1/k} )$ switches are necessary and that $O ( n^{1 + 1/k} ( \log n )^{1/k} )$ switches are sufficient for such networks.

Gossiping without Duplicate Transmissions

Douglas B. West

SIAM. J. on Algebraic and Discrete Methods 3, pp. 418-419 (2 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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$n$ people have distinct bits of information, which they communicate via telephone calls in which they transmit everything they know. We require that no one ever hear the same piece of information twice. In the case 4 divides $n,\,n\geqq 8$, we provide a construction that transmits all information using only $9n/4 - 6$ calls. Previous constructions used $\frac{1}{2}n \log n$ calls.

On the Computation of the Competition Number of a Graph

Robert J. Opsut

SIAM. J. on Algebraic and Discrete Methods 3, pp. 420-428 (9 pages) | Cited 10 times

Online Publication Date: July 17, 2006

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This paper examines the problem of recognizing competition graphs (niche overlap graphs), a notion introduced and studied extensively by Cohen [Food Webs and Niche Space, Princeton Univ. Press, Princeton, NJ, 1978]. Beginning with an acyclic digraph $F = ( V,A )$, define its competition graph $K (F ) = ( V,E)$ by $( x,y ) \in E$ if and only if there exists a $w$ such that $( x,w ) \in A$ and $( y,w ) \in A$. A graph, $G$, is a competition graph if there exists an $F$ such that $G = K ( F )$. Roberts [Lecture Notes in Mathematics 642, Springer-Verlag, New York, 1978, pp. 477–490] studied recognizing competition graphs and, equivalently, computing an arbitrary graphs competition number, $k( G )$. The competition number, which he showed to be well defined, is the smallest $k$ such that $G \cup I_k $ is a competition graph. In this paper we settle a question posed by Roberts and show that recognizing competition graphs is NP-complete by reducing it to $R$-CONTENT as defined by Orlin [Nederl. Akad. Wetensch. Proc. Ser. A, 80 (1977), pp. 406–424]. We also give bounds on $k ( G )$ in terms of $R$-Content $( G )$ and compute $k ( G )$ for the class of line graphs using a technique similar to that in Roberts.

Algorithms for Testing the Diagonal Similarity of Matrices and Related Problems

Gernot M. Engel and Hans Schneider

SIAM. J. on Algebraic and Discrete Methods 3, pp. 429-438 (10 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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A simple algorithm is presented for testing the diagonal similarity of two square matrices with entries in a field. Extended forms of the algorithm decide various related problems such as the simultaneous diagonal similarity of two families of matrices, the existence of a matrix in a subfield diagonally similar to a given matrix, the existence of a unitary matrix similar to a given complex matrix, and the corresponding problems for diagonal equivalence in place of diagonal similarity. The computational complexity of our principal algorithm is studied, programs and examples are given. The algorithms are based on the existence of a canonical form for diagonal similarity. In the first part of the paper theorems are proved which establish the existence of this form and which investigate its properties.

New Methods for Evaluating Distrebuation Automation and Control (DAC) Sysyem Benefits

John Peschon and Dale Ross

SIAM. J. on Algebraic and Discrete Methods 3, pp. 439-452 (14 pages)

Online Publication Date: July 17, 2006

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The decade ahead will be one of heightened concern for costs versus benefits to end users of electric energy. More than in the past, the distribution planner will be concerned about investment costs, operating efficiency and reliability of service. The advent of new dispersed generation, storage and control technologies for distribution systems will change not only the alternatives available to the planner, but also the planning methods themselves.
This paper summarizes the development of new distribution planning methods. In particular, methods have been developed for both expansion planning and operations planning of radial distribution systems. A particular application of distribution automation and control will be for temporary distribution system reconfiguration during either forced outages or maintenance/construction-related outages. Remotely controlled switches can be used to transfer load among radial feeders during construction, maintenance or other service interruptions—thereby reducing or preventing outages for many customers. This paper describes computerized methods for evaluating the reliability benefits of such advanced distribution systems.

Practical Applicatons of Discrete Mathmatical Programming in Exxon

William P. Drews

SIAM. J. on Algebraic and Discrete Methods 3, pp. 453-464 (12 pages)

Online Publication Date: July 17, 2006

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Applications of discrete mathematical programming may be subdivided into those involving economies of scale, those involving mutually exclusive variables and those involving nonconvexity in the constraint set. Exxon’s earliest successful applications involved investment planning under economies of scale. Operations scheduling applications are characterized by mutually exclusive variables: these have been solved satisfactorily by heuristic methods and by branch-and-bound methods running under stream-lined computational procedures. Nonconvex constraints are found in engineering design problems: these require artful formulation and specialized computational search procedures. Research is still needed to endow discrete mathematical programming with interactive computation capabilities, with enhanced analytical and interpretive options and with extensions into the domain of mathematical programming under uncertainty.

Sorting and Merging in Rounds

R. Häggkvist and P. Hell

SIAM. J. on Algebraic and Discrete Methods 3, pp. 465-473 (9 pages) | Cited 7 times

Online Publication Date: July 17, 2006

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The need for sorting algorithms which operate in a fixed number of rounds (rather than have each new comparison depend on the outcomes of all previous comparisons) arises in structural modeling. Since all comparisons within a round are evaluated simultaneously, such algorithms have an obvious connection to parallel processing.
In an earlier paper (SIAM J. Comput.,10 (1981), pp. 465–472) we used a counting argument to prove the existence of subquadratic sorting algorithms for two rounds. Here we develop optimal algorithms for merging in rounds, and apply them to actually construct good sorting algorithms for $k$ rounds, $k\geqq 3$. For example, in $k = 66$ rounds, our algorithm will sort any $n$-element linearly ordered set with $O ( n^{1.10} )$ comparisons.

The Bounded Path Tree Problem

Paolo M. Camerini and Giulia Galbiati

SIAM. J. on Algebraic and Discrete Methods 3, pp. 474-484 (11 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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The subject of this paper is the bounded path tree (BPT) problem: An undirected graph $G ( V,E )$ is given whose edges have nonnegative lengths; two subsets $I$ and $J$ of $V$ are also given, and nonnegative constants $U_i $, $W_i $ are associated with each $i \in I$, $j \in J$. The BPT problem asks for a tree of $G$ whose vertex set contains $I \cup J$ and whose path joining vertices $i$ and $j$ is not longer than $U_i + W_j $, for each $i \in I$, $j \in J$. This problem generalizes the shortest path and the minimum longest path spanning tree problem. It complements standard min–max location problems, as it asks for a tree given the facility locations, instead of locating facilities in a given network. In this paper we propose some applications of the BPT problem for the design of emergency and communication networks, show its equivalence to an extension of the absolute center location problem and give an algorithm for its solution. This algorithm requires time $O( k| E | + k | V | \log k )$, where $k = | I \cup J |$, plus time for finding in $G$ all shortest path lengths between a vertex in $I \cup J$ and a vertex in $V$. We also consider a few simple extensions of the BPT problem, such as those admitting negative or multiple edge lengths, lower (as well as upper) bounds to path lengths, constants $Z_{ij} $ instead of $U_i + W_i $. We show that all these extensions are NP-complete.

The Interval Count of a Graph

R. Leibowitz, S. F. Assmann, and G. W. Peck

SIAM. J. on Algebraic and Discrete Methods 3, pp. 485-494 (10 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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The interval count of an interval graph $G$ is the minimum number of different interval sizes needed to represent the vertices of $G$, where two vertices are adjacent if and only if their intervals intersect.
We show that if $G$ is an interval graph and for some vertex $x$, $G - \{ x \}$ has interval count one, then $G$ has interval count two or less.
We also show how to construct examples of interval graphs where the interval count of $G$ exceeds that of $G - \{ x \}$ by at least two when the latter number is two or more.

Majorization of Finite Partially Ordered Sets

Ko-Wei Lih

SIAM. J. on Algebraic and Discrete Methods 3, pp. 495-503 (9 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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The classical concept of majorization between two finite sequences of real numbers is extended to between real-valued functions defined on a finite partially ordered set. We establish characterizations of majorization. The FKG and Holley inequalities from statistical mechanics have their majorization interpretations. Their validity is closely tied to the distributiveness of the background lattice. An equivalence proof is hence provided. Finally, suitable restrictions on the rearrangement of function values enable us to generalize the classical theorem of Schur–Ostrowski which characterizes functions preserving majorization in terms of relative orders of their first partial derivatives.

A Primal Approach to the Simple Plant Location Problem

Gerard Cornuejols and Jean-Michel Thizy

SIAM. J. on Algebraic and Discrete Methods 3, pp. 504-510 (7 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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The most successful algorithms for solving simple plant location problems are presently dual-based procedures. However, primal procedures have distinct practical advantages (e.g., in sensitivity analysis). We propose a primal subgradient algorithm to solve the well-known strong linear programming relaxation of the problem. Typically this algorithm converges very fast to a point whose objective value is close to the integer optimum and where most of the decision variables have been fixed either to 0 or to 1. To fix the values of the remaining variables we use a greedy-interchange algorithm. Thus we propose thiss approach as a heuristic. Computational experience shows that an optimal solution is discovered with high frequency.

Vertices Belonging to All or to No Maximum Stable Sets of a Graph

P. L. Hammer, P. Hansen, and B. Simeone

SIAM. J. on Algebraic and Discrete Methods 3, pp. 511-522 (12 pages) | Cited 7 times

Online Publication Date: July 17, 2006

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The focus of the present paper is on the relations between the set $D$ of optimal solutions of a maximum weighted stable set problem, and the set $C$ of optimal solutions of its continuous relaxation. The main result is that if a variable takes a constant binary value in all$\hat X \in C$, then it takes the same value in all$\hat X \in D$ (this may be contrasted with a well-known result of Nemhauser and Trotter, stating that if a variable takes a binary value in some$\hat X \in C$, then it takes the same value in some$X \in D$). For any graph $G$, the set $P$ of the vertices $j$ such that $\hat X_j $ has a constant binary value in all$\hat X \in C$, can be efficiently detected; moreover, the results in this paper imply that in the unweighted case, the subgraph induced by $P$ has the “strong” König–Egerváry property and that the subgraph induced by the complement of $P$ has a perfect 2-matching: actually, the maximum stable sets of $G$ are in a 1-to-1 correspondence with those of the latter subgraph.

Minimizing a Combinatorial Function

Ding Zhu Du and F. K. Hwang

SIAM. J. on Algebraic and Discrete Methods 3, pp. 523-528 (6 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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Let $M( N,d )$ denote the minimax number of group tests required for the identification of the $d$ defectives in a set of $N$ items. It is of interest to determine the values of $N$ and $d$ for which $M ( N,d ) = N - 1$ (achieved by testing the first $N - 1$ items one by one). Recently it has been proved that $M ( N,d ) = N - 1$ for $N < \lfloor 2.5 d \rfloor $. A lemma crucially used to obtain that result is the following: \[M ( N,d )\geqq \min \left\{ N - 1,2 t + \left\lceil \log_2 \begin{pmatrix} N - t \\ d - t \end{pmatrix} \right\rceil \right\}. \] The problem is to find a suitable $t$ such that \[N - 1\leqq 2t + \left\lceil \log _2 \begin{pmatrix} N - t \\ d - t \end{pmatrix} \right\rceil \] and $d/N$ is minimized. However, standard methods do not work for this minimization problem. In this paper we propose a novel method to solve the minimization problem to obtain the new result: $M ( N,d ) = N - 1$ for $N\leqq \lfloor 2.625d \rfloor $.

Fixed Point Behavior of Threshold Functions on a Finite Set

Eric Goles

SIAM. J. on Algebraic and Discrete Methods 3, pp. 529-531 (3 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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In this paper we obtain a sufficient condition that a kind of iteration scheme has no cycles other than fixed points.
A detailed version of this result and of its applications may be found in E. Goles [Tech. Rep., Depto. Matem., Univ. de Chile, Santiago, 1981].

Speed-Up in Dynamic Programming

F. Frances Yao

SIAM. J. on Algebraic and Discrete Methods 3, pp. 532-540 (9 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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Dynamic programming is a general problem-solving method that has been used widely in many disciplines, including computer science. In this paper we present some recent results in the design of efficient dynamic programming algorithms. These results illustrate two approaches for achieving efficiency: the first by developing general techniques that are applicable to a broad class of problems, and the second by inventing clever algorithms that take advantage of individual situations.

An Algorithm for Partitioning the Nodes of a Graph

Earl R. Barnes

SIAM. J. on Algebraic and Discrete Methods 3, pp. 541-550 (10 pages) | Cited 39 times

Online Publication Date: July 17, 2006

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Let $G = \{ N,E \}$ be an undirected graph having nodes $N$ and edges $E$. We consider the problem of partitioning $N$ into $k$ disjoint subsets $N_1 , \cdots ,N_k $ of given sizes $m_1 , \cdots ,m_k $, respectively, in such a way that the number of edges in $E$ that connect different subsets is minimal. We obtain a heuristic solution from the solution of a linear programming transportation problem.

Finite Solution Theory for Coalitional Games

William F. Lucas and Kai Michaelis

SIAM. J. on Algebraic and Discrete Methods 3, pp. 551-565 (15 pages)

Online Publication Date: July 17, 2006

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In 1944 von Neumann and Morgenstern introduced a theory of solutions (stable sets) for multiperson cooperative games in characteristic function form. Some special classes of games are known to have solutions which are finite sets. These finite solutions give rise to interesting geometrical structures and basic combinatorial patterns. They have provided new insights into problems in the social sciences and they invite additional interpretations and uses in the mathematical and physical sciences. This paper provides an introduction and survey of finite solution theory.

Double Semiorders and Double Indifference Graphs

Margaret B. Cozzens and Fred S. Roberts

SIAM. J. on Algebraic and Discrete Methods 3, pp. 566-583 (18 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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The notion of semiorder was introduced by Luce in 1956 as a model for preference in the situation where indifference judgments are nontransitive. The notion of indifference graph was introduced by Roberts in 1968 as a model for nontransitive indifference. Motivated by problems of measurement and serration in the social sciences and by frequency assignment problems in communications, we discuss generalizations called double semiorders and double indifference graphs. Semiorders are exactly the binary relations $(A,P)$ such that there is a real-valued function $f$ on $A$ satisfying $xPy$ iff $f ( x ) > f ( y ) + \delta $, where $\delta$ is a fixed positive number. Indifference graphs are exactly the graphs $( V,E )$ such that there is a real-valued function $f$ on $V$ satisfying $\{ x,y \} \in E$ iff $| f ( x ) > f ( y ) | \leqq \delta $. Suppose $\delta _1 > \delta _2 > 0$. We present conditions on a pair of binary relations $( A,P_1 )$ and $( A,P_2 )$ necessary and sufficient for the existence of a real-valued function $f$ on $A$ satisfying $xP_i y$ iff $f ( x ) > f ( y ) + \delta _i $, $i = 1,2$. These lead to conditions on $( V,E_1 )$ and $( V,E_2 )$ necessary and sufficient for the existence of a real-valued function $f$ on $V$ satisfying $\{ x,y \} \in E_i $ iff $| f ( x ) - f ( y ) |\leqq \delta _i,\, i = 1,2$.

On the Greedy Heuristic for Continuous Covering and Packing Problems

Marshall L. Fisher and Laurence A. Wolsey

SIAM. J. on Algebraic and Discrete Methods 3, pp. 584-591 (8 pages) | Cited 11 times

Online Publication Date: July 17, 2006

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Worst-case bounds are given on the performance of the greedy heuristic for a continuous version of the set covering problem. This generalizes results of Chvatal, Johnson and Lovasz for the 0-1 covering problem. The results for the greedy heuristic and for other heuristics are obtained by treating the covering problem as a limiting case of a generalized location problem for which worst-case results are known. An alternative approach involving dual greedy heuristics leads also to worst-case bounds for continuous packing problems.

Recursive Algorithms for Unitary and Symplectic Group Representations

Kenneth Baclawski

SIAM. J. on Algebraic and Discrete Methods 3, pp. 592-605 (14 pages)

Online Publication Date: July 17, 2006

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The finite-dimensional irreducible representations of the unitary groups $U ( n )$, $SU( n )$ and the symplectic groups $Sp( 2n,\mathbb{C} )$ are explicitly constructed using recursive algorithms. A simple labelling system is described that provides a unique label for each vector in a specific basis of every irreducible representation, and the algorithms act in a “combinatorial” manner on those labels. The algorithms are examples of lexicographic straightening algorithms.
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