Showing papers in "Siam Journal on Algebraic and Discrete Methods in 1982"
TL;DR: In this article, it was shown that it is NP-hard to determine if a partial order has dimension 3, and several other related dimension-type problems are shown to be NP-complete.
Abstract: The dimension of a partial order P is the minimum number of linear orders whose intersection is P. There are efficient algorithms to test if a partial order has dimension 1 or 2. We prove that it is NP-complete to determine if a partial order has dimension 3. As a consequence, several other related dimension-type problems are shown to be NP-complete.
456 citations
TL;DR: In this paper, it was shown that each of the terms in the matrix W and U that corresponds to an enumerated forest occurs just once and all the other terms cancel, and the sign of each term is determined by the parity of the linking from U to W contained in the forest.
Abstract: Let $( A_{ij} )$, i, $j \in V$ be the matrix with entries $ - a_{ij} $ if $i
e j$ and diagonal entries such that all the column sums are zero. Let $ a_{ij} $ be a variable associated with arc $ij$ in the complete digraph G on vertices V. Let $| A ( \bar{W}|\bar{U} ) |$ be the matrix that results from deleting sets of k rows W and columns U from A. The all minors matrix tree theorem states that $ | A ( \bar{W}|\bar{U} ) |$ enumerates the forests in G that have (a) k trees, (b) each tree contains exactly one vertex in U and exactly one vertex in W, and (c) each arc is directed away from the vertex in U of the tree containing the arc. We give an elementary combinatorial proof in which we show that each of the terms in $| A ( \bar{W}|\bar{U} ) |$ that corresponds to an enumerated forest occurs just once and the other terms cancel. The sign of each term is determined by the parity of the linking from U to W contained in the forest, and is easy to calculate explicitly in the proof.The results are extended to ...
347 citations
TL;DR: A heuristic algorithm for partitioning the nodes of a graph into a given number of subsets in such a way that the number of edges connecting the various subsets is a minimum.
Abstract: 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
323 citations
TL;DR: In this article, the authors give new formulations and a new proof of Laman's theorem, based on matroid theory, and then apply these to prove the following result: if G is 6-connected, then it will be rigid in the plane.
Abstract: Let G be a graph. Let us place the points of G in “general” position in the plane and then replace its edges by rigid bars (with flexible joints). We would like to know if the resulting structure is rigid and if not, compute its “degree of freedom”. This problem was solved by Laman [6] (see also [2]). In this note we give some new formulations and a new proof of Laman’s theorem, based on matroid theory, and then apply these to prove the following result: if G is 6-connected, then it will be rigid in the plane. We also construct infinitely many 5-connected graphs which do not have this property.
272 citations
TL;DR: In this article, a general decomposition theory can be applied to the resulting digraph decomposition, a consequence is a theorem which asserts the uniqueness of a decomposition of any digraph, each member of the decomposition being either indecomposable or special.
Abstract: A composition for directed graphs which generalizes the substitution (or X-join) composition of graphs and digraphs, as well as the graph version of set-family composition, is described. It is proved that a general decomposition theory can be applied to the resulting digraph decomposition. A consequence is a theorem which asserts the uniqueness of a decomposition of any digraph, each member of the decomposition being either indecomposable or “special”. The special digraphs are completely characterized; they are members of a few interesting classes. Efficient decomposition algorithms are also presented.
270 citations
TL;DR: This work proposes a hybrid algorithm, based on algorithms for simpler bin packing problems, and shows that proof techniques developed for the simpler cases can be combined to prove close bounds on the worst case behavior of the new hybrid.
Abstract: Suppose we are given a set L of rectangular items and wish to pack them into identical rectangular bins, so that no two items overlap and so that the number of bins used is minimized. This generali...
226 citations
TL;DR: In this article, a new type of spread is constructed by combining geometric, group theoretic and matrix methods, which correspond to affine translation planes, and new Kerdock sets are obtained having various interesting properties.
Abstract: In an orthogonal vector space of type $\Omega ^ + ( 4n,q )$, a spread is a family of $q^{2n - 1} + 1$ totally singular $2n$-spaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a $2m$-dimensional vector space over $GF ( q )$, a spread is a family of $q^m + 1$ subspaces of dimension m which induces a partition of the points of the underlying projective space; these spreads correspond to affine translation planes. By combining geometric, group theoretic and matrix methods, new types of spreads are constructed and old examples are studied. New Kerdock sets and new translation planesare obtained having various interesting properties.
138 citations
TL;DR: This investigation considers the problem of nonpreemptively assigning a set of independent tasks to a system of identical processors to maximize the earliest processor finishing time and proves that the worst-case performance of the LPT algorithm has an asymptotically tight bound of $4}{3}$ times the optimal.
Abstract: This investigation considers the problem of nonpreemptively assigning a set of independent tasks to a system of identical processors to maximize the earliest processor finishing time. While this goal is a nonstandard scheduling criterion, it does have natural applications in certain maintenance scheduling and deterministic fleet sizing problems. The problem is NP-hard, justifying an analysis of heuristics such as the well-known LPT algorithm in an effort to guarantee near-optimal results. It is proved that the worst-case performance of the LPT algorithm has an asymptotically tight bound of $\frac{4}{3}$ times the optimal.
113 citations
TL;DR: In this article, it was shown that every planar graph can be partitioned into two or more components of roughly equal size by deleting only $O( \sqrt{n} )$ vertices, and such a partitioning can be found in O( n )$ time.
Abstract: The results in this paper are closely related to the effective use of the divide-and-conquer strategy for solving problems on planar graphs. It is shown that every planar graph can be partitioned into two or more components of roughly equal size by deleting only $O ( \sqrt{n} )$ vertices, and such a partitioning can be found in $O ( n )$ time. Some of the theorems proved in the paper are improvements on the previously known theorems while others are of more general form. An upper bound for the minimum size of the partitioning set is found.
112 citations
TL;DR: One of the fundamental problems of phase retrieval in spectroscopic analysis is of a combinatorial nature and can be solved using purely algebraic techniques as mentioned in this paper, given two sets A and B in some Euclidean space.
Abstract: One of the fundamental problems of phase retrieval in spectroscopic analysis is of a combinatorial nature and can be solved using purely algebraic techniques. Given two sets A and B in some Euclide...
89 citations
TL;DR: In this paper, the reliability polynomial associated with an independence system was shown to be Ω(g(p) = √ √ n f_k p^k ( 1 - p )^{n - k}, where f is the number of independent sets of cardinality k and n is the cardinality of ground set.
Abstract: The reliability polynomial associated with an independence system is $g ( p ) = \sum_{k = 0}^n f_k p^k ( 1 - p )^{n - k} $, where $f_k $ is the number of independent sets of cardinality k and n is the cardinality of the ground set. An independence system $( T,\Gamma )$ is shellable if all maximal independent sets have the same cardinality and if there exists an ordered partition of the set of independent sets into intervals $\{ [ F_i ,G_i ] \}_{i = 1}^I $ (an interval $ [ F ,G ] = \{ F^\prime :F \subseteq F^\prime \subseteq G \}$) where for all $n^\prime $, $n^\prime \leqq I$, $G_{n^\prime } $, is a maximal independent set and ($T, \cup _{i=1}^{n^\prime } [ F_i ,G_i ]$) is an independence system. For the class of shellable independence systems, tight upper and lower bounds are given on $g ( p )$, when the number of maximal independent sets and the number of minimum cardinality dependent sets are fixed. These results can be applied to obtain bounds on the reachability measure, which is the probability that...
TL;DR: In this article, it was shown that the problem of recognizing competition graphs is NP-hard and can be reduced to R-CONTENT as defined by Orlin [Nederl. Akad. Ser. this article.
Abstract: 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. ...
TL;DR: The results of this paper illustrate two approaches for achieving efficiency in the design of efficient dynamic programming algorithms: 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.
Abstract: 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.
TL;DR: In this paper, the authors studied the relation between the set D of optimal solutions of a maximum weighted stable set problem and the set C of optimal solution of its continuous relaxation problem.
Abstract: 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” Konig–Egervary 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.
TL;DR: A polynomial time algorithm is given that finds a min-cut linear arrangement of trees whose cost is within a factor of 2 of optimal.
Abstract: The min-cut linear arrangement problem is one of several one-dimensional layout problems for undirected graphs that may be of relevance to VSLI design.This paper gives a polynomial time algorithm that finds a min-cut linear arrangement of trees whose cost is within a factor of 2 of optimal. For complete m-ary trees a linear time algorithm is given that finds an optimum min-cut linear arrangement.
TL;DR: In this paper, it is shown that a permutationally convex game has a nonempty core and that both convex games and m.c.s.t. games are permutational convex.
Abstract: Notwithstanding the apparent differences between convex games and minimum cost spanning tree (m.c.s.t.) games, we show that there is a close relationship between these two types of games. This close relationship is realized with the introduction of the group of permutationally convex (p.c.) games. It is shown that a p.c. game has a nonempty core and that both convex games and m.c.s.t. games are permutationally convex.
TL;DR: The main result of as mentioned in this paper is that a chordal graph with $n$ vertices and $m$ edges can be cut in half by removing $O(sqrt{m}) vertices.
Abstract: Chordal graphs are undirected graphs in which every cycle of length at least four has a chord. They are sometimes called rigid circuit graphs or perfect elimination graphs; the last name reflects their utility in modelling Gaussian elimination on sparse matrices. The main result of this paper is that a chordal graph with $n$ vertices and $m$ edges can be cut in half by removing $O(\sqrt{m})$ vertices. A similar result holds if the vertices have non-negative weights and we want to bisect the graph by weight, or even if we want to bisect the graph simultaneously by several unrelated sets of weights.
TL;DR: In this paper, a necessary and sufficient condition for a ranked partially ordered set to be rank symmetric, rank unimodal and strongly Sperner is presented, which is used to provide a new short proof that this combination of properties is preserved under the product operation.
Abstract: A ranked partially ordered set is said to be Sperner if it has no antichain bigger than its largest rank. A necessary and sufficient condition for a ranked partially ordered set to be rank symmetric, rank unimodal and strongly Sperner is presented. This condition involves representations of $\mathfrak{sl} ( 2,\mathbb{C} )$. It is used to provide a new, short proof that this combination of properties is preserved under the product operation. The sufficient part of this condition is also used to provide new, simpler proofs that certain combinatorially interesting partially ordered sets are rank symmetric, rank unimodal and strongly Sperner.
TL;DR: This work is interested in the minimum possible number of switches in rearrangeable networks in which the depth is at most k and the length of the longest path from an input to an output is at least k.
Abstract: 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.
TL;DR: In this article, the class of mean residual life functions and sequences is characterized and the utility of such characterizations for modelling life distributions through empirically determined mean residual lives is discussed.
Abstract: The class of mean residual life functions and sequences is characterized. Apart from the utility of such characterizations for modelling life distributions through empirically determined mean residual lives, it is shown that such functions arise naturally in many areas such as branching processes. Several additional consequences regarding various nonparametric classes of life distributions are derived, including some characterizations of the exponential and uniform distributions.
TL;DR: In this paper, worst-case bounds on the performance of the greedy heuristic for a continuous version of the set covering problem were obtained for both the 0-1 and 1-covering problems.
Abstract: 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.
TL;DR: In this article, the authors obtained a sufficient condition that a kind of iteration scheme has no cycles other than fixed points, which is the same as the condition of the fixed point condition in this paper.
Abstract: 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].
TL;DR: In this paper, the chromatic number of Steiner triple systems of order v was shown to be at most Θ(n 2 ) for any ε > 0, where ε is the number of vertices connected by exactly one edge.
Abstract: In this paper, several results on the chromatic number of Steiner triple systems are established. A Steiner triple system is a simple 3-uniform hypergraph in which every pair of vertices is connected by exactly one 3-edge. Among other things, we prove that for any $k\geqq 3$ there exists an $n_k $ such that for all admissible $v \geqq n_k $ there exists a k-chromatic Steiner triple systems of order v. In addition we prove that for all $v \geqq 49$ there exists a 4-chromatic Steiner triple system of order v. An estimate of $n_k $ is also established, namely, $c_1 k^2 \log k > n_k > c_2 k^2 $.
TL;DR: In this article, double semi-orders and double indifference graphs were introduced as a model for preference in the situation where indifference judgments are nontransitive, and conditions on a pair of binary relations were presented for the existence of a real-valued fun.
Abstract: 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 fun...
TL;DR: This work develops optimal algorithms for merging in rounds, and applies them to actually construct good sorting algorithms for k rounds, which will sort any n-element linearly ordered set with n^{1.10} ) comparisons.
Abstract: 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.
TL;DR: In this paper, it was shown that G(P ) is perfect, i.e., the graph whose vertices are the maximal rectangles in P, two such vertices being adjacent if the corresponding rectangles have nontrivial intersection.
Abstract: Let P be a simply connected polyomino. Let $G( P )$ be the graph whose vertices are the maximal rectangles in P, two such vertices being adjacent if the corresponding rectangles have nontrivial intersection. In this paper we show that $G ( P )$ is perfect. This solves a problem posed by Berge et al.
TL;DR: In this paper, it was shown that any connected graph may be transformed by a sequence of switchings to any other connected graph of the same degree sequence, in such a way that all the intermediate graphs formed are connected.
Abstract: This paper follows as a natural extension of the ideas in a previous paper where we showed that any connected graph may be transformed by a sequence of switchings to any other connected graph of the same degree sequence, in such a way that all the intermediate graphs formed are connected. This was done for simple graphs, multigraphs and pseudographs. Here we show that the corresponding result is true for 2-connected simple graphs. The result for multigraphs and pseudographs will appear elsewhere. We also note that for k-connected graphs where $k\geqq 3$, this transformation theorem seems much more difficult to prove and in the last section of this paper we mention these difficulties.
TL;DR: In this paper, it was shown that many hookup classes of graphs are isomorphism complete, and gave polynomial isomorphisms for the others, including chordal graphs with bounded maximum clique size.
Abstract: Hookup classes are classes of graphs with a certain type of recursive definition, which can be viewed as a generalization of k-trees. We show that many hookup classes of graphs are isomorphism complete, and give polynomial isomorphism algorithms for the others. Other results in this paper include the development of a structural decomposition for hookup graphs and similar isomorphism results for generalizations of hookup classes, including a polynomial isomorphism testing algorithm for chordal graphs with bounded maximum clique size.
TL;DR: In this article, the authors investigated the exponent of a primitive, nearly reducible matrix A and showed that it is possible to find gaps in the exponent set of A with exponent k, where s is the length of a shortest circuit in the directed graph associated with A.
Abstract: A nonnegative matrix is called nearly reducible provided it is irreducible and the replacement of any positive entry by zero yields a reducible matrix. The purpose of this article is to investigate the exponent $\gamma ( A )$ of an $n \times n$ primitive, nearly reducible matrix A. Our principal result is that $\gamma ( A )\leqq n + s ( n - 3 )$, where s is the length of a shortest circuit in the directed graph associated with A. It is an easy application of this result to find gaps in the exponent set of $n \times n$ primitive, nearly reducible matrices. We also show that for integers n, k satisfying $n\geqq k - 1\geqq 5$ there exists an $n \times n$ primitive, nearly reducible matrix with exponent k. The proofs are carried out by means of directed graphs.
TL;DR: This work proposes a primal subgradient algorithm to solve the well-known strong linear programming relaxation of the problem, and shows that an optimal solution is discovered with high frequency.
Abstract: 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.