Showing papers in "Siam Journal on Algebraic and Discrete Methods in 1980"

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

Abstract: Techniques from algebraic geometry, in particular the hard Lefschetz theorem, are used to show that certain finite partially ordered sets $Q^X $ derived from a class of algebraic varieties X have the k-Sperner property for all k. This in effect means that there is a simple description of the cardinality of the largest subset of $Q^X $ containing no $( k + 1 )$-element chain. We analyze, in some detail, the case when $X = G/P$, where G is a complex semisimple algebraic group and P is a parabolic subgroup. In this case, $Q^X $ is defined in terms of the “Bruhat order” of the Weyl group of G. In particular, taking P to be a certain maximal parabolic subgroup of $G = SO( 2n + 1 )$, we deduce the following conjecture of Erdos and Moser: Let S be a set of $2\ell + 1$ distinct real numbers, and let $T_1 , \cdots ,T_k $ be subsets of S whose element sums are all equal. Then k does not exceed the middle coefficient of the polynomial $2( 1 + q )^2 ( 1 + q^2 )^2 \cdots ( 1 + q^\ell )^2 $, and this bound is best poss...

The Complexity of Coloring Circular Arcs and Chords

Abstract: The word problem for products of symmetric groups, the circular arc graph coloring problem, and the circle graph coloring problem, as well as several related problems, are proved to be $NP$-complete. For any fixed number K of colors, the problem of determining whether a given circular arc graph is K-colorable is shown to be solvable in polynomial time.

On Additive Bases and Harmonious Graphs

Abstract: This paper first considers several types of additive bases. A typical problem is to find $n_\gamma ( k )$, the largest n for which there exists a set $\{ 0 = a_1 < a_2 < \cdots < a_k \}$ of distinct integers modulo n such that each r in the range $0\leqq r \leqq n - 1$ can be written at least once as $r \equiv a_i + a_j $ (modulo n) with $i < j$. For example, $n_\gamma ( 8 ) = 24$, as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or $i < j$ to $i\leqq j$, or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels, $ \cdots $) are shown to be harmonious while others (even cycles, most complete or complete...

The Elimination Matrix: Some Lemmas and Applications

Abstract: Two transformation matrices are introduced, L and D, which contain zero and unit elements only. If A is an arbitrary $( n,n )$ matrix, L eliminates from vecA the supradiagonal elements of A, while D performs the inverse transformation for symmetricA. Many properties of L and D are derived, in particular in relation to Kronecker products. The usefulness of the two matrices is demonstrated in three areas of mathematical statistics and matrix algebra: maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.

The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities

Abstract: Let ${\mathbf{T}}$ denote the transition matrix of an ergodic chain, $\mathbf{\mathcal{C}}$ , and let ${\mathbf{A}} = {\mathbf{I}} - {\mathbf{T}}$. Let ${\mathbf{E}}$ be a perturbation matrix such that $\mathbf{\tilde{T}} = {\mathbf{T}} - {\mathbf{E}}$ is also the transition matrix of an ergodic chain, $\mathbf{\tilde{\mathcal{C}}}$. Let $\boldsymbol{\omega} $ and $\boldsymbol{\tilde \omega} $ denote the limiting probability (row) vectors for $\mathbf{\mathcal{C}}$ and $\mathbf{\tilde{\mathcal{C}}}$. The purpose of this paper is to exhibit inequalities bounding the relative error $\| \boldsymbol{\omega} - \boldsymbol{\tilde \omega} \| / \| \boldsymbol{\omega} \|$ by a very simple function of ${\mathbf{E}}$ and ${\mathbf{A}}$. Furthermore, the inequality will be shown to be the best one which is possible. This bound can be significant in the numerical determination of the limiting probabilities for an ergodic chain.In addition to presenting a sharp bound for $\| \boldsymbol{\omega} - \boldsymbol{\tilde \om...

Dynamic-Programming Algorithms for Recognizing Small-Bandwidth Graphs in Polynomial Time

Abstract: In this paper we investigate the problem of testing the bandwidth of a graph: Given a graph, G, can the vertices of G be mapped to distinct positive integers so that no edge of G has its endpoints mapped to integers which differ by more than some fixed constant, k? We exhibit an algorithm to solve this problem in $O ( f ( k )N^{k + 1} )$ time, where N is the number of vertices of G and $f ( k )$ depends only on k This result implies that the “Bandwidth $\overset{?}{\leqq} k$” problem is not NP-complete (unless P = NP) for any fixed k, answering an open question of Garey, Graham, Johnson, and Knuth We also show how the algorithm can be modified to solve some other problems closely related to the “Bandwidth $\overset{?}{\leqq} k$” problem

Extremal Values of the Interval Number of a Graph

Abstract: The interval number $i( G )$ of a simple graph G is the smallest number t such that to each vertex in G there can be assigned a collection of at most t finite closed intervals on the real line so that there is an edge between vertices v and w in G if and only if some interval for v intersects some interval for w. The well known interval graphs are precisely those graphs G with $i ( G )\leqq 1$. We prove here that for any graph G with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil $. This bound is attained by every regular graph of degree d with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil $ for graphs on n vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor $ for graphs with e edges.

Worst-Case Analysis of Network Design Problem Heuristics

Abstract: The Optimal Network problem (as defined by A. J. Scott, The optimal network problem : Some computational procedures, Trans. Res., Vol 3 (1969) pp. 201–210) consists of selecting a subset of arcs that minimizes the sum of the shortest paths between all nodes subject to a budget constraint. This paper considers the worst-case behavior of heuristics for this problem. Let n be the number of nodes in the network and $\varepsilon $ be a constant between 0 and 1. For a general class of Optimal Network Problems, we show that the question of finding a solution which is always less than $n^{1 - \varepsilon } $ times the optimal solution is $NP$-complete. This indicates that all polynomial-time heuristics for the problem most probably have poor worst-case performance. An upper bound for worst-case heuristic performance of 2n times the optimal solution is also derived. For a restricted version of the Optimal Network problem we describe a procedure whose maximum percentage of error is bounded by a constant.

On the Structure of t-Designs

Abstract: It is possible to view the combinatorial structures known as (integral) t-designs as $\mathbb{Z}$-modules in a natural way. In this note we introduce a polynomial associated to each such $\mathbb{Z}$-module. Using this association, we quickly derive explicit bases for the important class of submodules which correspond to the so-called null-designs.

Single Facility $l_p $-Distance Minimax Location

Abstract: We discuss the problem of locating a new facility among n given demand points on a plane. The maximum weighted distance to demand points must be minimized. The general $l_p $-norm $( p\geqq 1)$ is used as distance measure. The method is quite fast computationally: for example, a 3000 demand point problem in $l_p $ is solved in half a second.

A Canonical Representation of Simple Plant Location-problems and its Applications

Abstract: We consider a location problem whose mathematical formulation is $\max_S \{ v( S ): S \subseteq N,|S| = K \}$ where $v ( S ) = v ( \emptyset ) + \sum_{i \in I} \max_{j \in S} c_{ij}, C = ( c_{ij} )$ is a real matrix with row index set $I = \{ 1, \cdots ,m \}$ and column index set $N = \{ 1, \cdots ,n \}$ and $K\leqq n$ is a positive integer A set function of the form $v ( S )$ is called a simple location function We give a constructive proof that a set function w is a simple location function if and only if it can be represented in the canonical form $w ( S ) = r_\emptyset + \Sigma _{T \cap S e \emptyset } r_T $ with $r_T \geqq 0$ for all $\emptyset \subset T \subset N$ The proof is also a polynomial algorithm for reducing the matrix C to the canonical form We give two applications of this canonical representation The first is that a large class of algorithms for the location problem need to enumerate all but l of the feasible solutions in order to find one of the lth best solutions The second app

The Growth of Powers of a Nonnegative Matrix

Abstract: Let A be a nonnegative $n \times n$ matrix. In this paper we study the growth of the powers $A^m, m = 1,2,3, \cdots $ when $\rho ( A ) = 1$. These powers occur naturally in the iteration process \[x^{( m + 1 )} = Ax^{( m )} ,\quad x^{( 0 )} \geqq 0,\] which is important in applications and numerical techniques. Roughly speaking, we analyze the asymptotic behavior of each entry of $A^m $. We apply our main result to determine necessary and sufficient conditions for the convergence to the spectral radius of A of certain ratios naturally associated with the iteration above.

The Erdös-Ko-Rado Theorem for Integer Sequences

Abstract: For positive integers $n,k,t$ we investigate the problem how many integer sequences $( a_1 ,a_2 , \cdots ,a_n )$ we can take, such that $1\leqq a_i \leqq k$ for $1\leqq i\leqq n$, and any two sequences agree in at least t positions. This problem was solved by Kleitman (J. Combin. Theory, 1 (1966), pp. 209–214) for $k = 2$, and by Berge (in Hypergraph Seminar, Columbus, Ohio (1972), Springer-Verlag, New York, 1974) for $t = 1$. We prove that for $t\geqq 15$ the maximum number of such sequences is $k^{n - t} $ if and only if $k\geqq t + 1$.

The FKG Inequality and Some Monotonicity Properties of Partial Orders

Abstract: Let $( a_1 , \cdots ,a_m ,b_1 , \cdots ,b_n )$ be a random permutation of $1,2, \cdots ,m + n$. Let P be a partial order on the a’s and b’s involving only inequalities of the form $a_i < a_j $ or $b_i < b_j $, and let $P'$ be an extension of P to include inequalities of the form $a_i < b_j$; i.e, $P' = P \cup P''$, where $P''$ involves only inequalities of the form $a_i < b_j $. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular (*) $\Pr ( a_1 < b_1 |P' ) \geq \Pr ( a_1 < b_1 |P )$. We give a simple example to show that the more general inequality (*) where P is allowed to contain inequalities of the form $a_i < b_j $ is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality.

On the Order of Random Channel Networks

Abstract: The order of a stream with no tributaries is defined to be 1. In general, when two streams of orders $\alpha $ and $\beta $ flow together, the larger stream thus produced has order $\max \{ \alpha ...

A Combinatorial Problem Arising in the Study of Reaction-Diffusion Equations

Abstract: We study a discrete model based on the observed behavior of excitable media. This model has the basic properties of an excitable medium, that is, a threshold phenomenon, at refractory period, and a...

A Characterization of Weighted Arithmetic Means

Abstract: We prove, among other things, that the set of weighted arithmetic means is identical with the set of functions $f:R^n \to R$ satisfying (i) $\min \{ x_j \}\leqq f ( x_1 ,x_2 , \cdots ,x_n )\leqq \max \{ x_j \}$ and (ii) for $k = 2,3:\sum _{i = 1}^k x_{ij} = s( j = 1,2, \cdots ,n ) \Rightarrow \sum _{i = 1}^k f( x_{i1} ,x_{i2} , \cdots ,x_{in} ) = s$.

A Phase-Type Semi-Markov Point Process

Abstract: A semi-Markov point process is defined for which the intervals of time between successive events have phase-type distribution. The distribution of the number of events in an interval is examined, and it is shown how the expected number of events in an interval may be efficiently computed. A stationary version of the process is analyzed. In particular, a necessary and sufficient condition, and simple sufficient conditions under which the new process is a renewal process, are determined.

Random Colorings of a Lattice of Squares in the Plane

Abstract: Square cells, tesselating the plane in a lattice arrangement, will be colored black or white by a random process. The coloring tries to imitate the appearance of cells with statistically independent colors, with black and white equally likely. Here only a relatively small initial set of cells is colored independently; the remaining colors are then determined by solving a linear recurrence equation. In this way one obtains colorings which, for some value of n, have independent colors in every set of n cells. The value of n, which depends on the recurrence equation used, can be deduced from divisibility properties of certain polynomials.

Dilworth Numbers, Incidence Maps and Product Partial Orders

Abstract: The Dilworth numbers of a finite partially ordered set $P,\{ d_k ( P )|k\geqq 0 \}$, are the maximum sizes of the union of k antichains. We give a characterization of the Dilworth numbers in terms of the dimensions of the kernels of certain linear maps on the complex vector space generated by the elements of P. This is applied to prove bounds on the Dilworth numbers of product partial orders in terms of those of its factors and to prove sufficient conditions for the product of two partial orders to have the Sperner property.

Some Erdös–Ko–Rado Theorems for Chevalley Groups

Abstract: For each infinite family of Chevalley groups over a finite field an Erdos–Ko–Rado theorem is given. The technique uses orthogonal polynomials to find upper bounds for the independence number of specific graphs. In all but one case the bound is realizable.

On the Average Shape of Binary Trees

Abstract: The average level numbers of the leaves of a binary tree are studied, where each binary tree is regarded as being equally likely. A formula is derived for the number of binary trees with jth leaf at a prescribed level. The asymptotic behavior of the average level number of the jth leaf is determined. The average level numbers are shown to first increase and then decrease.

Decomposing a Permutation into Two Large Cycles: An Enumeration

Abstract: Let $c_{l,m,\tau }^{( n )} $ denote the number of ways a permutation $\tau $ can be expressed as the product of an l-cycle and an m-cycle, all in the symmetric group on n symbols. In 1972, the first author gave a necessary and sufficient condition on l such that $c_{l,i,\tau }^{( n )} > 0$ for every even permutation $\tau $. In 1978, G. Boccara gave a necessary and sufficient condition on $l,m,$ and $\tau $ such that $c_{l,m,\tau }^{( n )} > 0$. More recently, D. W. Walkup developed a recursion for $c_{n,n,\tau }^{( n )} $. In this paper, we show how to recursively calculate the values of $c_{n,n - i,\tau }^{( n )} $. Theorem 1 states that $c_{n,n - 1,\tau }^{( n )} = 2 \cdot ( n - 2 )$! for every odd $\tau $. Theorem 2 exhibits $c_{n + 1,n - i,\sigma}^{( n + 1 )} $ as a linear, combination (with easily obtained integral coefficients) of a specified set of $c_{n,n - i,\tau }^{( n )} $. Applications include a method to evaluate, by inverting an integral triangular matrix, all values in {$c_{n,n,\tau }^{( n...

Orthogonal Polynomials in Two Variables of q-Hahn and q-Jacobi Type

Abstract: Two families of orthogonal polynomials in two discrete variables are constructed for a weight function of q-hypergeometric type. The polynomials are expressed in terms of q-Hahn polynomials. The connection coefficients between the two families involve Askey and Wilson’s ${}_4 \phi _3 $-polynomials, which are certain balanced, terminating basic hypergeometric series. The method is to consider functions on the lattice of subspaces of a finite vector space which are invariant under the subgroup of the corresponding general linear group which fixes a pair of nested subspaces.By limiting methods, corresponding results are obtained for Andrews and Askey’s little q-Jacobi polynomials, which are orthogonal on a countable compact set. The classical Hahn version of the theory, where the underlying group is the symmetric group, has been worked out by the author in a previous paper.

On Unimodality for Linear Extensions of Partial Orders

Abstract: R. Rivest has recently proposed the following intriguing conjecture: Let $x^* $ denote an arbitrary fixed element in an n-element partially ordered set P, and for each k in $\{ 1,2, \cdots ,n \}$ l...

Candidate Keys and Antichains

Abstract: It is shown that a matrix can be constructed in which a row is determined when specified in any collection of column sets with the property that $B \subset C, A \subset B$ implies $A \subset C$. The problem of determining the minimum number of rows needed for a collection C on n rows in worst case is raised. Some bounds are given.

An $O ( ( n\log p )^2 )$ Algorithm for the Continuous p-Center Problem on a Tree

Abstract: This paper considers the problem of locating p facilities on a tree network in order to minimize the maximum of the distances of the points on the network to their respective nearest facilities. An $O ( ( n\log p )^2 )$ algorithm for a tree network with n nodes is presented.

A Proof of Tuite’s Trinity Theorem and a New Determinant Formula

Abstract: A new proof of Tutte’s trinity theorem (Proc. Cambridge Phil. Soc., 1948), (North-Holland, 1973) is presented. The proof is based on a new determinant formula for the number of spanning arborescences of a digraph. This formula generalizes the determinant formula given by Maurer (SIAM J. Appl. Math., 1976) for the number of spanning trees of an undirected graph.

Totally Nonnegative, M-, and Jacobi Matrices

Abstract: It is shown among other results that a nonsingular M-matrix is a Jacobi matrix if and only if its inverse is totally nonnegative and it is a normal Jacobi matrix if and only if its inverse is oscillatory.This is an extension of a previous result of Markham [Proc. Amer. Math. Soc., 161 (1912), pp. 326–330].

A Note on the NP-Completeness of Vertex Elimination on Directed Graphs

Abstract: A correction is made to Rose and Tarjan’s proof [SIAM J. Appl. Math., 1978] that determining a minimum fill-in elimination ordering for a directed graph is NP-complete.