Showing papers in "SIAM Journal on Discrete Mathematics in 1992"
TL;DR: Given a simple graph G and a positive number d, an L_d ( 2,1 )-labelling of G is a function f:V ( G ) to [ 0,fty ) such that whenever x,y are adjacent, f ( x ) - f ( ...
Abstract: Given a simple graph $G = ( V,E )$ and a positive number d, an $L_d ( 2,1 )$-labelling of G is a function $f:V ( G ) \to [ 0,\infty )$ such that whenever $x,y \in V$ are adjacent, $| f ( x ) - f ( ...
701 citations
TL;DR: It is shown that, for inputs of length n, the probabilistic (bounded error) communication complexity of set intersection is $\Theta ( n )$.
Abstract: It is shown that, for inputs of length n, the probabilistic (bounded error) communication complexity of set intersection is $\Theta ( n )$. Since set intersection can be recognized nondeterministic...
600 citations
TL;DR: An algorithm for computing the edge-connectivity of a multigraph, where p ( \leqq | E | )$ is the number of pairs of nodes between which G has an edge, is proposed and consists only of times of graph searches and edge contractions.
Abstract: Given an undirected graph $G = ( V,E )$, it is known that its edge-connectivity $\lambda ( G )$ can be computed by solving $O( | V | )$ max-flow problems The best time bounds known for the problem
413 citations
TL;DR: A min-max formula is derived for $\gamma$ and a polynomial time algorithm to compute it is described, and the directed counterpart of the problem is solved and is shown to be NP-complete.
Abstract: What is the minimum number $\gamma$ of edges to be added to a given graph G so that in the resulting graph the edge-connectivity between every pair $\{ u,v \}$ of its nodes is at least a prescribed value $r( u,v )$?Generalizing earlier results of S. Sridhar and R. Chandrasekaran [Integer Programming and Combinatorial Optimization, R. Kannan and W. Pulleyblank, eds., Proceedings of a conference held at the University of Waterloo, University of Waterloo Press, Waterloo, Ontario, Canada, 1990, pp. 467–484] (when G is the empty graph), of K. P. Eswaran and R. E. Tarjan [SIAM Journal on Computing, 5 (1976), pp. 653–665] (when $r ( u,v ) \equiv 2$), and of G.-R. Cai and Y.-G. Sun [Networks, 19 (1989 ), pp. 151–172 ] (when $r ( u,v ) \equiv k\geqq 2$, we derive a min-max formula for $\gamma$ and describe a polynomial time algorithm to compute $\gamma$. The directed counterpart of the problem is solved in the same sense for the case when $r ( u,v ) \equiv k\geqq 1$ and is shown to be NP-complete if $r ( u,v ) \eq...
320 citations
TL;DR: An axiomatic characterization of the position value is provided and relations with the Myerson value are discussed, and, for special classes of communication situations, elegant calculation methods for their position values are described.
Abstract: A new solution concept for communication situations is considered: the position value. This concept is based on an evaluation of the importance of the various communication links between the players. An axiomatic characterization of the position value is provided for the class of communication situations where the communication graphs contain no cycles. Furthermore, relations with the Myerson value are discussed,and, for special classes of communication situations, elegant calculation methods for their position values are described.
263 citations
TL;DR: In this chapter, the structure of the graph is completely encoded, so that, given the labels of two vertices, one can test if they are adjacent in time linear in the size of the labels.
Abstract: How to represent a graph in memory is a fundamental data structuring question. In the usual representations of an n-vertex graph, the names of the vertices (i.e., integers from 1 to n) betray nothing about the graph itself. Indeed, the names (or labels) on the n vertices are just $\log n$ bit place holders to allow data on the edges to encode the structure of the graph. In this scenario, there is no such waste. By assigning $O(\log n)$ bit labels to the vertices, the structure of the graph is completely encoded, so that, given the labels of two vertices, one can test if they are adjacent in time linear in the size of the labels. Furthermore, given an arbitrary original labeling of the vertices, structure coding labels are found (as above) that are no more than a small constant factor larger than the original labels. These notions are intimately related to vertex-induced universal graphs of polynomial size. For example, planar graphs can be labeled with structure coding labels of size $ < 4\log n$, which i...
255 citations
TL;DR: This paper attaches a name to a natural class of combinatorial problems and points out that the class includes many important special cases, and one special case is shown to be NP-complete.
Abstract: This paper attaches a name to a natural class of combinatorial problems and points out that the class includes many important special cases. One special case, a simple problem of placing nonoverlapping labels on a rectangular map, is shown to be NP-complete.
203 citations
TL;DR: A family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information and guarantee a stretch factor of O( k^2 ) on the length of the routes.
Abstract: This paper presents a family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information. The quality of the routes provided by a scheme is measured in terms of their stretch, namely, the maximum ratio between the length of a route connecting some pair of processors and their distance. The hierarchical schemes$\mathcal{H}_k $ (for every integer $k\geqq 1$) presented in this paper guarantee a stretch factor of $O( k^2 )$ on the length of the routes and require storing at most $O( k \cdot n^{1/k} \cdot \log n\log D )$ bits of routing information per vertex in an n-processor network with diameter D. The schemes are name independent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k.The proposed schemes are based on a new efficient solution to a certain graph-theoretic problem concerning sparse graph covers. The new cover technique has already found several other a...
197 citations
TL;DR: The relative powers of queues and stacks are compared as mechanisms for laying out the edges of a graph in a k-queue layout, and each edge is assigned to exactly one of the k queues so that the edges assigned to each stack obey a last-in/first-out discipline.
Abstract: The relative powers of queues and stacks are compared as mechanisms for laying out the edges of a graph. In a k-queue layout, vertices of the graph are placed in some linear order (also called a linear arrangement), and each edge is assigned to exactly one of the k queues, so that the edges assigned to each queue obey a first-in/first-out (FIFO) discipline. As the vertices are scanned left to right, an edge is enqueued on its assigned queue when its left endpoint is encountered and is dequeued from its queue when its right endpoint is encountered. In a k-stack layout, vertices of the graph are placed in some linear order, and each edge is assigned to exactly one of the k stacks so that the edges assigned to each stack obey a last-in/first-out discipline. As the vertices are scanned left to right, an edge is pushed on its assigned stack when its left endpoint is encountered and is popped from its stack when its right endpoint is encountered. The paper has three main results. First, a tradeoff between queue...
158 citations
TL;DR: Each of two parties holds an n-bit input, x and y, respectively and they wish to privately compute the value of f( x,y) , which should not be known to the other party.
Abstract: Each of two parties $P_\mathcal{X} $ and $P_\matcal{Y} $ holds an n-bit input, x and y, respectively. They wish to privately compute the value of $f( x,y )$. That is, $P_\mathcal{X} $ should not le...
146 citations
TL;DR: It is conjectured that the roots of the reliability polynomial of any connected graph lie in the disc $| z - 1 | \leq 1$, and evidence for this conjecture is provided.
Abstract: The reliability of a graph G is the probability that G is connected, given that edges are independently operational with probability p. This is known to be a polynomial in p, and the location of the roots of these functions is discussed. In particular, it is conjectured that the roots of the reliability polynomial of any connected graph lie in the disc $| z - 1 | \leq 1$, and evidence for this conjecture is provided. It is shown that all real roots lie in $\{ 0 \} \cup ( 1,2 ]$ and that every graph has a subdivision for which the roots of the reliability polynomial lie in the conjectured disc.
TL;DR: The existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems is nonconstructively proved as discussed by the authors.
Abstract: The existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems is nonconstructively proved. Some were not previously known to be in $\mathcal{P}$ at all; others were only known to be in $\mathcal{P}$ by way of brute force or dynamic programming formulations with unboundedly high-degree polynomial running times. The methods applied include the recent Robertson–Seymour theorems on the well-partial-ordering of graphs under both the minor and immersion orders. The complexity of search versions of these problems is also briefly addressed.
TL;DR: This paper presents polynomially bounded algorithms for finding a cycle through any two prescribed arcs in a semicomplete digraph and for findingA cycle throughAny two prescribed vertices in a complete k-partite oriented graph.
Abstract: This paper presents polynomially bounded algorithms for finding a cycle through any two prescribed arcs in a semicomplete digraph and for finding a cycle through any two prescribed vertices in a complete k-partite oriented graph. It is also shown that the problem of finding a maximum transitive subtournament of a tournament and the problem of finding a cycle through a prescribed arc set in a tournament are both NP-complete.
TL;DR: Lower bounds are given and bounded degree graphs are constructed that allow rapid broadcasting in graphs with fixed maximum degree in which broadcasting can be completed in near minimum time.
Abstract: Broadcasting is an information dissemination process in which a message is to be sent from a single originator to all members of a network by placing calls over the communication lines of the network. Several previous papers have investigated methods to construct sparse graphs (networks) in which this process can be completed in minimum time from any originator. The graphs produced by these methods contain high degree vertices. [Liestman and Peters, SIAM Journal on Discrete Mathematics, 1 (1988), pp. 531–540 ] and [Bermond and Peyrat, Proceedings of the 19th SE Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, 1988, pp. 283–292] began an investigation of graphs with fixed maximum degree in which broadcasting can be completed in near minimum time. This investigation is continued in this paper by giving lower bounds and constructing bounded degree graphs that allow rapid broadcasting. The constructions use ideas developed by Jerrum and Skyum [IEEE Transactions on Computers, C-...
TL;DR: This paper considers “black box groups,” i.e., finite groups whose elements are uniquely encoded by strings of uniform length, with group operations being performed by a group oracleB.
Abstract: This paper considers “black box groups,” i.e., finite groups whose elements are uniquely encoded by strings of uniform length, with group operations being performed by a group oracleB. Let G, H be such groups, each given by a list of generators. It is known that the problem of membership in G belongs to ${\text{NP}}^B $ [L. Babai and E. Szemeredi, Proceedings of the 25th IEEE Symposium on the Foundation of Computer Science, 1984, pp. 229–240]. The following problems are shown to belong to the complexity class ${\text{AM}}^B $; i.e., they possess bounded-round randomized interactive proofs (Arthur–Merlin protocols): nonmembership in G, the verification of the order of G, isomorphism of G and H, and checking the list of composition factors of G. A group oracle B is constructed, under which none of these problems belongs to ${\text{NP}}^B $, even for abelian groups.All the results extend to “black box factor groups,” i.e., groups defined as factor groups $G/N$, where G is a black box group, $N \triangleleft ...
TL;DR: A derivation in a transformational system such as a graph grammar may be redundant in the sense that the exact order of the transformations may not affect the final outcome; all that matters is that each transformation, when applied, is applied to the correct substructure.
Abstract: A derivation in a transformational system such as a graph grammar may be redundant in the sense that the exact order of the transformations may not affect the final outcome; all that matters is that each transformation, when applied, is applied to the correct substructure. By taking advantage of this redundancy, we can develop an efficient encoding scheme for such derivations. This encoding scheme has a number of diverse applications. It can be used in efficient enumeration of combinatorial objects or for compact representation of program and data structure transformations. It can also be used to derive lower bounds on lengths of derivations. It is shown, for example, that $\Omega ( n \log n )$ applications of the associative and commutative laws are required in the worst case to transform an n-variable expression over a binary associative, commutative operation into some other equivalent expression. Similarly, it is shown that $\Omega ( n\log n )$ “diagonal flips” are required in the worst case to transf...
TL;DR: The orders of these poles define timescales associated with the process when $\varepsilon $ is small, and an algorithm is developed for computing these orders.
Abstract: Let $P ( 0 ) \in R^{n \times n} $ be a stochastic matrix representing transition probabilities in a Markov Chain. Also, for a matrix $A \in R^{n \times n} $ whose row-sums are zero, let $P( \varepsilon ) \equiv P ( 0 ) + \varepsilon A$ be stochastic and irreducible for all $0 < \varepsilon \leq \varepsilon _{\max } $, for some $\varepsilon_{\max } $. Finally, let $M( \varepsilon )$ be a matrix whose $( i, j )$ entry is the mean passage time from state i to state j when transitions are governed by $P( \varepsilon )$. When the Markov chain associated with $P( 0 )$ is decomposable into a number of independent chains plus a set of transient states, some of the entries of $M( \varepsilon )$ have singularities at zero. The orders of these poles define timescales associated with the process when $\varepsilon $ is small. An algorithm is developed for computing these orders. The only input required is the supports of $P( 0 )$ and A, making the problem a combinatorial one. Finally, it is shown how the orders of the...
TL;DR: A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given and several known lower bound on cover times to be systematically proved.
Abstract: A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given. This technique is used to prove (i) A tight bound of $\Omega ( | V |\log^2 | V | )$ for the two-dimensional torus; (ii) A tight bound of $\Omega ( | V |\log ^2 | V |/ \log d_{\max } )$ for trees with maximum degree $d_{\max } $; (iii) Tight bounds of $\Omega ( \mu ^ + \log ^2 | V | )$ for rapidly mixing walks on vertex transitive graphs, where $\mu^+$ denotes the maximum expected hitting time between vertices.In addition to these new results, the technique allows several known lower bounds on cover times to be systematically proved, often in a much simpler way.Finally, a different technique is used to prove an $\Omega ( 1 / ( 1 - \lambda _2 ) )$ lower bound on the cover time, where $\lambda_2 $ is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [...
TL;DR: It is proved that all finite grid graphs of positive width have Hamiltonian line graphs, and sufficient conditions for a grid graph to be Hamiltonian.
Abstract: This paper presents sufficient conditions for a grid graph to be Hamiltonian. It is proved that all finite grid graphs of positive width have Hamiltonian line graphs.
TL;DR: A new lower bound for unconstrained quadratic 0 – 1 minimization can be computed by solving a linear programming problem of polynomial size in the number of variables; and it is shown that the polyhedron defined by the constraints of this LP formulation is precisely the first Chvatal closure of thepolyhedron associated with standard linearization procedures.
Abstract: In this paper a new lower bound for unconstrained quadratic 0 – 1 minimization is investigated It is shown that this bound can be computed by solving a linear programming problem of polynomial size in the number of variables; and it is shown that the polyhedron ${\text{S}}^{[3]} $, defined by the constraints of this LP formulation is precisely the first Chvatal closure of the polyhedron associated with standard linearization procedures By rewriting the quadratic minimization problem as a balancing problem in a weighted signed graph, it can be seen that the polyhedron defined by the odd cycle inequalities is equivalent, in a certain sense, with ${\text{S}}^{[3]} $ As a corollary, a compact linear programming formulation is presented for the maximum cut problem for the case of weakly bipartite graphs
TL;DR: This paper presents a $\Theta ( n \log n )$ time and$ space algorithm for finding a maximum two-chain in a point set $\rho $, where n = | \rho |$.
Abstract: Given a point set $\rho $, a chain is a subset $\mathcal{C} \subseteq \rho $ of points in which, for any two points, one is dominated by the other. A two-chain is a subset of $\rho $ that can be partitioned into two chains. A two-chain with maximum cardinality among all possible two-chains is called a maximum two-chain. This paper presents a $\Theta ( n \log n )$ time and $\Theta ( n )$ space algorithm for finding a maximum two-chain in a point set $\rho $, where $n = | \rho |$. Maximum two-chain has applications in, for example, graph-theoretic problems, VLSI layout, and sequence manipulation.
TL;DR: A linear time algorithm for the case of 3-colored graphs is presented and it is shown that this problem in its general form is NP-complete and can be solved in $O( n^{k + 1} )$ time, where n is the number of vertices and kThe number of colors.
Abstract: The problem of determining whether a vertex-colored graph can be triangulated without introducing edges between vertices of the same color is what is of interest here. This problem is known to be polynomially equivalent to a fundamental problem in numerical taxonomy called the perfect phylogeny problem, which is concerned with the inference of evolutionary history. This problem is also related to the problem of recognizing partial k-trees, a class of graphs that has received much attention recently. The problem in its general form is NP-complete and can be solved in $O( n^{k + 1} )$ time, where n is the number of vertices and k the number of colors. In this paper, a linear time algorithm for the case of 3-colored graphs is presented.
TL;DR: It is proved that the edges of G can be covered by circuits whose total length is at most $m + ( r/r - 1 )( n - 1)$, where r is the minimum length of an even circuit of length at least 6.
Abstract: Let G be a bridgeless graph with m edges and n vertices. It is proved that the edges of G can be covered by circuits whose total length is at most $m + ( r/r - 1 )( n - 1 )$, where r is the minimum length of an even circuit (of G) of length at least 6 ($r = \infty $, if there is no such circuit). The proof suggests a polynomial-time algorithm for constructing such a cover.
[...]
TL;DR: The paper addresses the question of which complete bipartite graphs are 2-competition graphs, formulated as the following combinatorial problem: given disjoint sets A and B, when can one find n subsets of A tO B so that every a in A and b in B are together contained in at least two of the subsets.
Abstract: IfD (V, A) is a digraph, its p-competition graph for p a positive integer has vertex set V and an edge between x and y ifand only if there are distinct vertices a, , an in D with (x, a and (y, a) arcs ofD for each 1, , p. This notion generalizes the notion of ordinary competition graph, which has been widely studied and is the special case wherep 1. Results about the case wherep 2 are obtained. In particular, the paper addresses the question ofwhich complete bipartite graphs are 2-competition graphs. This problem is formulated as the following combinatorial problem: Given disjoint setsA and B such that A tO BI n, when can one find n subsets ofA tO B so that every a in A and b in B are together contained in at least two of the subsets and so that the intersection of every pair of subsets contains at most one element from A and at most one element from B?
TL;DR: A variant of the well-known gossip problem is studied and efficient schemes of calls are shown that assure complete communication with probability converging to 1 as n grows.
Abstract: A variant of the well-known gossip problem is studied. Each of n members of a communication network has a piece of information that should be made known to everybody else. This is to be done by placing a sequence of two-party phone calls along the lines of the network. During each call, the two participants exchange all information they currently have, in a unit of time. It is assumed that calls fail independently with fixed probability $0 < p < 1$ and that no information is exchanged during a failed call. For communication networks of bounded degree, efficient schemes of calls are shown that assure complete communication with probability converging to 1 as n grows. Both the number of calls and the time they use are of minimal order.
TL;DR: The low connectivity case is formulated on directed graphs, and the directed formulation is shown to give a better LP-relaxation than the undirected one, and it is shown how facet-defining inequalities of $P ( G )$ give facet- defining inequalities in this case.
Abstract: In this paper the problem of finding a minimum weight equivalent subgraph of a directed graph is considered. The associated equivalent subgraph polyhedron $P ( G )$ is studied. Several families of facet-defining inequalities are described for this polyhedron. A related problem of designing networks that satisfy certain survivability conditions, as introduced in [M. Grotschel and C. L. Monma, SIAM Journal on Discrete Mathematics, 3 (1990), pp. 502–523] is also studied. The low connectivity case is formulated on directed graphs, and the directed formulation is shown to give a better LP-relaxation than the undirected one. It is shown how facet-defining inequalities of $P ( G )$ give facet-defining inequalities in this case. Computational results are presented for some randomly generated problems.
TL;DR: A class of infinite graphs that can be embedded uniformly in the hyperbolic plane and carry nonconstant harmonic functions with finite Dirichlet sum is exhibited.
Abstract: A class of infinite graphs that can be embedded uniformly in the hyperbolic plane and carry nonconstant harmonic functions with finite Dirichlet sum is exhibited. In fact, a general method of constructing such harmonic functions “with prescribed boundary values” is provided.
TL;DR: This paper proves that if a Poset Q has an even number of linear extensions and these extensions can be generated by adjacent transpositions, then linear extensions of union of poset Q and an arbitrary poset P can also be generatedBy adjacent transposition.
Abstract: This paper proves that if a poset Q has an even number of linear extensions and these extensions can be generated by adjacent transpositions, then linear extensions of union of poset Q and an arbitrary poset P can also be generated by adjacent transpositions. This result is then applied to posets P and Q, which are sums of disjoint chains.
TL;DR: Consider a network in which n processors are connected by communication lines and are allowed to communicate with at most one other processor at a time, and an algorithm for constructing networks supporting the broadcast scheme and having approximately the minimum possible number of lines is given.
Abstract: Consider a network in which n processors are connected by communication lines and are allowed to communicate with at most one other processor at a time. Broadcast is the task of transmitting a message originated at one node to all other nodes in the network. Presented in this paper is a broadcasting scheme that can tolerate up to $k\leqq \lfloor \log \,n \rfloor $ line failures; that is, it assures that each node in the network will receive the message from the originator when up to $k \leqq \lfloor \log n \rfloor $ lines are faulty. The time required by the broadcast protocol is minimum, except in some cases that might require one unit of time more than the minimum. Moreover, an algorithm for constructing networks supporting the broadcast scheme and having approximately the minimum possible number of lines is given.
TL;DR: In this article, the existence of vertex disjoint paths in a directed planar graph is studied, and necessary and sufficient conditions are given for a directed path path that is vertex-disjoint whenever the vertices of the path are vertex disconnected.
Abstract: Let $D = ( V,A )$ be a directed planar graph, let $( r_1 ,s_1 ), \cdots , ( r_k ,s_k )$ be pairs of vertices on the boundary of the unbounded face, let $A_1 , \cdots ,A_k $ be subsets of A, and let H be a collection of unordered pairs from $\{ 1, \cdots ,k \}$. Given are necessary and sufficient conditions for the existence of a directed $r_i - s_i $ path $P_i $ in $( V,A_i )$ (for $i = 1, \cdots ,k$), such that $P_i $ and $P_j $ are vertex-disjoint whenever $\{ i, j \} \in H$.