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26
Optimum communication spanning trees
 SIAM J. Comput
, 1974
"... Abstract. Given a set of nodes N (i 1, 2,..., n) which may represent cities and a set of requirements ria which may represent the number of telephone calls between N and N j, the problem is to build a spanning tree connecting these n nodes such that the total cost of communication of the spanning tr ..."
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Cited by 74 (1 self)
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Abstract. Given a set of nodes N (i 1, 2,..., n) which may represent cities and a set of requirements ria which may represent the number of telephone calls between N and N j, the problem is to build a spanning tree connecting these n nodes such that the total cost of communication of the spanning tree is a minimum among all spanning trees. The cost of communication for a pair of nodes is r;a multiplied by the sum of the distances of arcs which form the unique path connecting Ni and N in the spanning tree. Summing over all () pairs of nodes, we have the total cost of communication of the spanning tree. Note that the problem is different from the minimum spanning tree problem solved by Kruskal and Prim. Key words, communication spanning trees, cuttree
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
Improved Algorithms For Bipartite Network Flow
, 1994
"... In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jE ..."
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Cited by 44 (7 self)
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In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a twoedge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the twoedge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...
A Faster Algorithm for Finding the Minimum Cut in a Directed Graph
 JOURNAL OF ALGORITHMS
, 1994
"... We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut sepa ..."
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Cited by 31 (0 self)
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We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n 2 maximum flow problems. We then show how to reduce the running time of these 2n 2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log(n 2 /m)) for finding the minimum cut in either a directed or an undirected network. © 1994 Academic Press, Inc. 1.
Every Graph With A Positive Cheeger Constant Contains A Tree With A Positive Cheeger Constant
 Funct. Anal
, 1997
"... . It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T (k) that has Cheeger constant k , but removing any edge from it reduces the Cheeger constant. Thi ..."
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Cited by 18 (2 self)
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. It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T (k) that has Cheeger constant k , but removing any edge from it reduces the Cheeger constant. This minimal graph, T (k) , is a tree, and every graph G with Cheeger constant h(G) ? k has a spanning forest in which each component is isomorphic to T (k) . 1. Introduction Let G be an infinite, locally finite graph; that is, every vertex has finitely many neighbors. The Cheeger constant of G is h(G) = inf K j@Kj jKj ; where K is any nonempty finite subset of V (G) , the set of vertices in G , and @K , the boundary of K , consists of all vertices in V (G) \Gamma K that have a neighbor in K . 1.1. Theorem. Let G be a (locally finite) graph with positive Cheeger constant. Then G contains a tree with positive Cheeger constant. J. Friedman and N. Pippenger [7] have shown that for positive ...
The VertexDisjoint Menger Problem In Planar Graphs
 SIAM J. Comput
, 1997
"... . We consider the problem of finding a maximum collection of vertexdisjoint paths in undirected, planar graphs from a vertex s to a vertex t. This problem is usually solved using flow techniques, which lead to O(nk) and O(n # n) running times, respectively, where n is the number of vertices and ..."
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Cited by 15 (4 self)
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. We consider the problem of finding a maximum collection of vertexdisjoint paths in undirected, planar graphs from a vertex s to a vertex t. This problem is usually solved using flow techniques, which lead to O(nk) and O(n # n) running times, respectively, where n is the number of vertices and k the maximum number of vertexdisjoint (s, t)paths. The best previously known algorithm is based on a divideandconquer approach and has running time O(n log n). The approach presented here is completely di#erent from these methods and yields a lineartime algorithm. Key words. graph algorithms, disjoint paths, planar graphs AMS subject classifications. 08C85, 68Q20, 05C38, 68R10, 90C35 PII. S0097539793253565 1. Introduction. The general Menger problem is a classical problem in both structural and algorithmic graph theory [2, 9, 15]. In general, it consists in finding the maximum number of vertexdisjoint or edgedisjoint paths in a graph from some designated vertex to another one. Th...
The rst eigenvalue of the Laplacian, isoperimetric constants, and the max ow mincut theorem
 Arch. Math
, 2006
"... Abstract. We show how ’test ’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical M ..."
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Cited by 8 (0 self)
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Abstract. We show how ’test ’ vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger’s constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger’s constant in terms of the inradius of a plane domain. 1.
A T_Xapproach to some results on cuts and metrics
 Advances in Applied Mathematics 19
, 1997
"... We give simple algorithmic proofs of some theorems of Papernov (1976) and Karzanov (1985,1990) on the packing of metrics by cuts. 1. ..."
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Cited by 6 (0 self)
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We give simple algorithmic proofs of some theorems of Papernov (1976) and Karzanov (1985,1990) on the packing of metrics by cuts. 1.
Chordless paths through three vertices
, 2006
"... Consider the following problem, which we call “Chordless path through three vertices ” or CP3V, for short: Given a simple undirected graph G = (V, E), a positive integer k, and three distinct vertices s, t, and v ∈ V, is there a chordless path of length at most k from s via v to t in G? In a chordle ..."
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Cited by 5 (0 self)
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Consider the following problem, which we call “Chordless path through three vertices ” or CP3V, for short: Given a simple undirected graph G = (V, E), a positive integer k, and three distinct vertices s, t, and v ∈ V, is there a chordless path of length at most k from s via v to t in G? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has arisen in the context of service deployment in communication networks. We resolve the parametric complexity of CP3V by proving it W [1]complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths and cycles. In particular, deciding on the existence of a single directed chordless (s, t)path in a digraph is also W [1]complete with respect to the length of the path.