Showing papers in "Siam Journal on Algebraic and Discrete Methods in 1986"
TL;DR: In this paper, the Minkowski-Krein-Milman theorem, Caratheodory's theorem and Tietze's convexity theorem were studied in graphs and hypergraphs.
Abstract: We study several notions of abstract convexity in graphs and hypergraphs. In each case, we obtain analogues of several classical results, including the Minkowski–Krein–Milman theorem, Caratheodory’s theorem and Tietze’s convexity theorem. In addition, our results yield new characterizations of the classes of chordal gaphs, strongly chordal graphs, Ptolemaic graphs and totally balanced hypergraphs.
280 citations
TL;DR: In this paper, the problem of modifying n-person games so as to take account of the difficulties imposed by lack of communications, and the opportunities this might accord to intermediaries is considered.
Abstract: We consider the problem of modifying n-person games so as to take account of the difficulties imposed by lack of communications, and the opportunities this might accord to intermediaries.In this model, the members of a finite set are simultaneously players in a game and vertices of a graph. A combination of these two structures gives rise to a new, modified game in which the only effective coalitions are those corresponding to connected partial graphs. We study the relationship between the power indices of the original game and the restricted game; for the special case where the graph is a tree, this relationship is especially easy to analyze.Several examples are studied in detail.
216 citations
TL;DR: A set of confluent graph reductions is found such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree.
Abstract: Our interest in the class of k-trees and their partial graphs and subgraphs is motivated by some practical questions about the reliability of communication networks in the presence of constrained line- and site-failures, and about the complexity of queries in a data base system. We have found a set of confluent graph reductions such that any graph can be reduced to the empty graph if and only if it is a subgraph of a 3-tree. This set of reductions yields a polynomial time algorithm for deciding if a given graph is a partial 3-tree and for finding one of its embeddings in a 3-tree when such an embedding exists. Our result generalizes a previously known recognition algorithm for partial 2-trees (series-parallel graphs).
185 citations
TL;DR: It is shown that the Bandwidth Minimization problem remains NP-complete even when restricted to “caterpillars with hairs of length at most three”, and the bandwidth problem is NP- complete whenrestricted to caterpillar with at most one hair attached to each vertex of the body.
Abstract: It is shown that the Bandwidth Minimization problem remains NP-complete even when restricted to “caterpillars with hairs of length at most three”. “Caterpillars” are special trees; they consist of a simple chain (the “body”) with various simple chains attached to thee vertices of the body (the attached chains are called “hairs”). A previous result in the literature shows that the bandwidth of caterpillars with hairs of length at most 2 can be found in $O( n\log n )$ time (this Journal, 2 (1981), pp. 387–393). We also show that the bandwidth problem is NP-complete when restricted to caterpillars with at most one hair attached to each vertex of the body. The proof is relatively straightforward and thereby also provides an easier proof than found in (SIAM J. Appl. Math., 34 (1978), pp. 477–495) that the bandwidth problem is NP-complete for trees with maximum vertex degree 3.
157 citations
TL;DR: In this paper, a number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant, including the enumeration of vertices and facets of a polytope, acyclic orientations of a graph and satisfying assignments of implicative boolean formulas.
Abstract: A number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant. Among others this is established for the numeration of vertices and of facets of a polytope, acyclic orientations of a graph and satisfying assignments of implicative boolean formulas.
138 citations
TL;DR: It is proved that the related problem of finding a sparsest null basis with an embedded identity matrix is NP-hard too and the zero–nonzero structure of sparsEST null bases is studied.
Abstract: The Null Space Problem (NSP) is the following: Given a $t \times n$ matrix A with $t < n$, find a sparsest basis for its null space (a null basis). We show that columns in a sparsest null basis correspond to minimal dependent sets of columns of A. Sparsest null bases are characterized by a greedy algorithm that augments a partial basis by a sparsest null vector. Despite this result, (NSP) is NP-hard since finding a sparsest null vector of A is NP-complete. We prove that the related problem of finding a sparsest null basis with an embedded identity matrix is NP-hard too. Finally, we study the zero–nonzero structure of sparsest null bases.
138 citations
TL;DR: The aim is to show that the Fredman–Komlos lemma is a special case of a simple inequality between entropies of graphs that enables us to handle more problems on separating partition systems.
Abstract: Fredman and Komlos have applied an interesting information-theoretic lemma to two problems in combinatorics. They have derived good lower bounds on the minimum size of a family of partitions of an n-element set into at most b classes such that all the subsets (respectively, pairs of subsets) of a certain kind are “separated” by at least one partition in the family.Our aim is to show that the Fredman–Komlos lemma is a special case of a simple inequality between entropies of graphs. The, general inequality enables us to handle more problems on separating partition systems. Part of the problems relate to hashing.
117 citations
TL;DR: In this paper, the problem of finding the numberings which minimize the edgesum of a given number of vertices in a graph has been shown to be NP-hard in the special case where the graph is the $2^n $ cube and for several instances of graphs with high degrees of symmetry.
Abstract: Given a numbering of the vertices of a graph, one can define the edgesum [6] as the sum of differences between adjacent vertices. The problem of finding numberings which are optimal in the sense of minimizing the edgesum is NP-complete [2] but has been solved in the special case where the graph is the $2^n $ cube [3] and for several instances of graphs with high degrees of symmetry [6]. We find the solutions for numberings of an $N \times N$ array. These have practical application in the problem of representing spatial information in a one-dimensional medium. To find our solutions, we exploit the fact that such numberings can always be taken to be ordered, in the sense that numbers increase along rows and down columns. We also consider a generalization of this problem to the case where the differences are raised to a power q. We derive bounds on the edgesum in this case, and show that the optimal numberings for $q < 1$ must be essentially different from those we have found for $q = 1$. While the latter ma...
116 citations
TL;DR: In this paper, the spectral radius of a complementary acyclic matrix A of 0's and l's was shown to be the largest root of the polynomial (n - 2 ) λ(n - 3 ) -1.
Abstract: For an $n \times n$ complementary acyclic matrix A of 0’s and l’s we show that the spectral radius $\rho ( A )$ of A satisfies $\rho ( A )\geqq n - 2$ and determine those matrices A for which equality holds. When A is an $n \times n$ irreducible, complementary tree matrix, we also obtain that $\rho ( A )\leqq \rho _n $, where $\rho _n $ is the largest root of the polynomial $\lambda^3 - (n - 2 )\lambda^2 - ( n - 3 )\lambda - 1$.
114 citations
TL;DR: In this paper, the maximal independent sets of vertices that any tree of n vertices can have were shown to have maximal number of maximal independent vertices, where vertices are independent sets.
Abstract: We find the largest number of maximal independent sets of vertices that any tree of n vertices can have.
104 citations
TL;DR: For an n-state finite, homogeneous, ergodic Markov chain, with transition matrix and stationary distribution, the authors assume that the entries of the transition matrix are differentiabl...
Abstract: For an n-state finite, homogeneous, ergodic Markov chain, with transition matrix ${\bf P}$ and stationary distribution ${\boldsymbol \pi} $ we assume that the entries of ${\bf P}$ are differentiabl...
TL;DR: This paper develops a complexity analysis of the general problem and derive a roundoff error bound on the Hessian approximation, and presents a concise characterization of methods known as substitution methods in graph-theoretic terms.
Abstract: Numerical optimization algorithms often require the (symmetric) matrix of second derivatives, $
abla ^2 f( x )$. If the Hessian matrix is large and sparse, then estimation by finite differences can be quite attractive since several schemes allow for estimation in much fewer than n gradient evaluations.The purpose of this paper is to analyze, from a combinatorial point of view, a class of methods known as substitution methods. We present a concise characterization of such methods in graph-theoretic terms. Using this characterization, we develop a complexity analysis of the general problem and derive a roundoff error bound on the Hessian approximation. Moreover, the graph model immediately reveals procedures to effect the substitution process optimally (i.e. using fewest possible substitutions given the differencing directions) in space proportional to the number of nonzeros in the Hessian matrix.
TL;DR: In this article, a generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions, and applications are given to Ulam's problem on longest increasing subsequences and to a law of large numbers for representations.
Abstract: Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and central indecomposable measures are investigated. A generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions. Applications are given to Ulam’s problem on longest increasing subsequences and to a law of large numbers for representations. An analogous theory for other graphs is discussed.
TL;DR: In this article, a closed-form solution with one summation over $n/k$ terms was obtained for consecutive k-out-of-n systems, where k = 2.
Abstract: Reliabilities for consecutive-k-out-of-n systems are typically given in the form of recursive equations. Some attempts have been made to use combinatorics to obtain closed-form solutions, but the solutions contain $k - 1$ summations. In this paper we obtain closed form solutions with one summation over $n/ k$ terms. For $k = 2$ we are able to eliminate all summations. We apply our result to compute the reliability of a k-loop computer network.
TL;DR: A parallel nonlinear Gauss–Seidel algorithm for approximating the solution of Au + \phi ( u ) = f where A is an M-matrix is introduced and studied and the speed-up on the Denelcor HEP parallel processing computer is recorded.
Abstract: Multi-splittings of a matrix are used to generate parallel algorithms to approximate the solutions of nonlinear algebraic systems. A parallel nonlinear Gauss–Seidel algorithm for approximating the solution of $Au + \phi ( u ) = f$ where A is an M-matrix is introduced and studied. Also, a parallel Newton–SOR method is defined for the problem $F ( u ) = 0$ where $F' ( u ) = $ the Jacobian is an M-matrix. An illustration and comparison of these methods with their serial versions is given. The speed-up on the Denelcor HEP parallel processing computer is also recorded.
TL;DR: A new generalized parity function is defined, and it is used to obtain a product construction of single-error-correcting codes (binary or not).
Abstract: We define a new generalized parity function, and use it to obtain a product construction of single-error-correcting codes (binary or not).
TL;DR: In this paper, a multigraph G on n vertices whose edges are linearly ordered is considered, and the vertices of G may represent people and the edges two-way communication between pairs of people.
Abstract: Consider a multigraph G on n vertices whose edges are linearly ordered. The vertices of G may represent people and the edges two-way communication between pairs of people. A vertex $\upsilon $ is k...
TL;DR: In this paper, the authors consider the problem of finding a spanning subgraph in a complete bipartite graph, i.e., a spanning subset of the vertices of the graph, each of whose components is a member of the set of stars.
Abstract: Given any set $\mathcal{B}$ of complete bipartite graphs, we ask whether a graph H admits a $\mathcal{B}$-factor, i.e., a spanning subgraph, each of whose components is a member of $\mathcal{B}$. More generally, we seek in H a maximum $\mathcal{B}$-packing, i.e., a $\mathcal{B}$-factor of a maximum size subgraph of H. We first treat the interesting special case when $\mathcal{B}$ is a set of stars. The results are generalized to arbitrary $\mathcal{B}$ in the last section. We prove for most of these problems that they are $\mathcal{N}\mathcal{P}$-hard; we also show that the remaining problems admit polynomial algorithms based on augmenting configurations. The simplicity of these algorithms, as well as the implied min-max theorems, resemble the theory of matchings in bipartite, rather than general, graphs.
TL;DR: This paper considers an important subclass of DAEs which can be solved by backward differentiation methods if their index does not exceed two, and presents an algorithm that determines if the index is one, two, or greater, based only on the structure.
Abstract: The index of many differential/algebraic equations (DAEs) is determined by the structure of the system, that is, by the pattern of nonzero entries in the Jacobians. This paper considers an important subclass of DAEs which can be solved by backward differentiation methods if their index does not exceed two. For this reason, it is desirable to determine whether the index exceeds two or not. In this paper we present an algorithm that determines if the index is one, two, or greater, based only on the structure. The algorithm can be exponential in its execution time: we do not know whether it is possible to get an asymptotically faster algorithm. However, in many practical problems, this algorithm will execute in polynomial time.
TL;DR: This paper provides a linear time algorithm to find the shortest simple paths from a given vertex to all other vertices and builds an algorithm to solve the uncapacitated plant location problem, where n is the number of vertices.
Abstract: It is well known that a series-parallel multigraph G can be constructed recursively from its edges. This construction is represented by a binary decomposition tree. This is a rooted binary tree T in which each vertex q corresponds to some series-parallel submultigraph of G, denoted by $G ( q )$, obtained as follows. Each leaf (tip) of T represents a distinct edge of G. If q is not a leaf then it is either of a series or parallel type. If $q_1 $ and $q_2 $ are the two sons of q on T, $G ( q )$ is the submultigraph obtained from $G ( q_1 )$ and $G ( q_2 )$ by the respective series or parallel composition.In this paper we use this tree to develop efficient algorithms for several optimization and selection problems defined on graphs with no $K_4 $ homeomorph. In particular, we provide a linear time algorithm to find the shortest simple paths from a given vertex to all other vertices. We also construct an $O ( n^4 )$ algorithm to solve the uncapacitated plant location problem, where n is the number of vertices...
TL;DR: For nonsingular n-by-n matrices A, the authors investigated the map A to \Phi (A ) \equiv A \circ ( A − 1} )^T, in which $ \circ $ denotes the Hadamard (entry-wise) product.
Abstract: For nonsingular n-by-n matrices A, we investigate the map \[ A \to \Phi ( A ) \equiv A \circ ( A^{ - 1} )^T \] in which $ \circ $ denotes the Hadamard (entry-wise) product. The matrix $\Phi ( A )$ arises in mathematical control theory in chemical engineering design problems, where it is known as the relative gain array, and also in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. We first give several elementary properties of $\Phi $ and show that the iterates $\Phi ^k ( A )$ converge to I when A is either positive definite or an H-matrix. We then discuss, with examples and partial results, several unsolved problems associated with $\Phi $. These include the range of $\Phi $, inverse images of elements in the range of $\Phi $, fixed points of $\Phi $, etc.
TL;DR: The decision problem associated with the problem of finding a point with largest norm in a bounded polyhedral set is shown to have a considerable range of complexity depending on the norm employed.
Abstract: The decision problem associated with the problem of finding a point with largest norm in a bounded polyhedral set is shown to have a considerable range of complexity depending on the norm employed. For a p-norm with integer $p\geqq 1$, the problem is shown to be NP-complete. For the $\infty $-norm, the problem can be solved in polynomial time. The problem of finding an upper bound to the largest norm for any $p \in [ 1,\infty ]$ can be solved in polynomial time by solving a single linear program.
TL;DR: In this article, it was shown that the spanning tree number t of a connected multigraph has a unique factorization, t = t_1 t_2 \cdots t_m, such that t_i is a multiple of t,i = 1,2, m -1,m -1$ and such that for every Abelian group A the group $B ( A )$ of bicycles over A is isomorphic to
Abstract: Let G be a connected multigraph and let $( A, + ,0 )$ be any Abelian group. For k an integer, let $A ( k )$ denote the subgroup of A given by $A ( k ) = \{ a \in A | ka = 0 \}$. A bicycle over A is a cycle over A that is also a cocycle. The set $B ( A )$ of bicycles over A determines a group. In this paper we show that the spanning tree number t of G has a unique factorization $t = t_1 t_2 \cdots t_m $ such that $t_i $ is a multiple of $t_{i + 1} ,i = 1,2, \cdots ,m - 1$ and such that for every Abelian group A the group $B ( A )$ of bicycles over A is isomorphic to $A ( t_1 ) \times A ( t_2 ) \times \cdots \times A ( t_m )$. Using this result we obtain a number of results on the spanning tree number including two formulae for the spanning tree number.
TL;DR: It is shown that a special class of circulants, known as Harary graphs, achieve this lower bound for all these values of i, and the necessary and sufficient conditions for a circulant to be super-$\lambda are determined.
Abstract: The connection between line-connectivity concepts of graphs and indices of network reliability is well-known. Of particular interest in such studies are the circulant graphs because the connected ones have the largest possible value of line-connectivity $\lambda $ of p-point, degree r, regular graphs, namely $\lambda = r$. In this work, we define the higher order line-connectivity measure $N_i $ as the number of line-disconnecting sets of order i. Regular degree r, p-point graphs having $\lambda = r$ satisfy $N_\lambda \geqq p$. Such graphs which attain this lower bound are called super-$\lambda $. In this work we determine the necessary and sufficient conditions for a circulant to be super-$\lambda$. In addition we determine a lower bound on $N_i $ for $\lambda \leqq i\leqq 2r - 3$. It is shown that a special class of circulants, known as Harary graphs, achieve this lower bound for all these values of i.
TL;DR: In this article, the Gauss-Seidel method and its variations constitute powerful tools for computing stationary distribution vectors for large-scale Markov process, such as those arising in queueing network analysis.
Abstract: Classical iterative schemes such as the Gauss–Seidel method and its variations constitute powerful tools for computing stationary distribution vectors for large-scale Markov process, such as those arising in queueing network analysis The coefficient matrix A in these processes in a Q-matrix, ie, a singular irreducible M-matrix with zero column sums and, unlike the nonsingular case, the classical iterations for A do not always converge The purpose of this paper is to survey the recent literature and to analyze the behavior of these methods completely in terms of the graph structure of A The results given here hold under somewhat weaker assumptions on A
TL;DR: In this article, the authors give an efficient algorithm to color the vertices of an outerplanar graph with the minimum number of colors, based on systematic coloring of elements (vertices and edges, respectively) of adjacent faces.
Abstract: The problems of finding values of the chromatic number and the chromatic index of a graph are NP-hard even for some restricted classes of graphs. Every outerplanar graph has an associated tree structure which facilitates algorithmic treatment. Using that structure, we give an efficient algorithm to color the vertices of an outerplanar graph with the minimum number of colors. We also establish algorithmically the value of the chromatic index of an outerplanar graph. Our algorithms are based on systematic coloring of elements (vertices and edges, respectively) of adjacent faces.
TL;DR: A short proof for the theorem that in a rectilinear, simply connected art gallery, $[ n /4 ]$ watchmen are sufficient, where n is the number of corners, in order for at least one to have a view of each internal point is given in this article.
Abstract: A short proof is given for the theorem that in a rectilinear, simply connected art gallery, $[ n /4 ]$ watchmen are sufficient, where n is the number of the corners, in order for at least one to have a view of each internal point.
TL;DR: In this article, the authors show asymptotic upper bounds on the strength of algebraically defined classes of degree k diffusers and show that these bounds can be achieved by certai...
Abstract: Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples of diffusers are defined by means of the action of elements of the matrix group $SL (2,{\bf Z} )$ on certain finite mathematical objects. Some corresponding, explicit expanders were introduced by Margulis. However, Gabber and Galil were the first to obtain good estimates for the expanders and produce from them a family of directed acyclic superconcentrators having density 271.8. In this paper we review various techniques for making expanders from diffusers. We also demonstrate asymptotic upper bounds on the strength of algebraically defined classes of degree k diffusers. Each upper bound is given as the norm of a diffusion operator on an infinite discrete group, and bounds for several examples are calculated. Numerical evidence is supplied in support of our conjecture that these bounds can be achieved by certai...
TL;DR: In this paper, singular value decomposition and perturbation analysis are applied to the Jacobian of robot kinematics; the condition number of the Jacobians is then proposed to be a measure of the "nearness" to degeneracy.
Abstract: In designing and evaluating industrial robots, it is important to find optimal configurations and locate optimum points in the workspace for the anticipated tasks. In the current paper the singular value decomposition and perturbation analysis are applied to the Jacobian of robot kinematics; the condition number of the Jacobian is then proposed to be a measure of the “nearness” to degeneracy. Then qualitative measures called kinematic “manipulability” and “sensitivity” are proposed. Some properties of proposed measures are investigated and the relation between these measures are discussed. Optimal postures of various types of industrial robots are obtained.
TL;DR: Algorithms and software for the solution of large sparse systems of linear algebraic equations with emphasis on systems arising in the numerical solution of partial differential equations are developed.
Abstract: We are interested in the development of algorithms, based on iterative methods, and software for the solution of large sparse systems of linear algebraic equations with emphasis on systems arising in the numerical solution of partial differential equations. The objective is to develop algorithms and software which are effective when used with a vector computer such as the Control Data CYBER 205 or the CRAY 1. A package of programs, known as ITPACK, has been developed for use on conventional, or scalar machines. A number of “short-range” modifications to ITPACK, including changes in the data storage format and changes in the programming, but not in the algorithms used, have been made and tested on a number of numerical examples. Preliminary work is described on “long-range” modifications which will involve extensive changes in the basic algorithms in order to achieve efficient vectorization.