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267
Linear network coding
 IEEE Trans. Inform. Theory
, 2003
"... Abstract — With network coding, intermediate nodes between the source and the receivers of an endtoend communication session are not only capable of relaying and replicating data messages, but also of coding incoming messages to produce coded outgoing ones. Recent studies have shown that network c ..."
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Cited by 582 (14 self)
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Abstract — With network coding, intermediate nodes between the source and the receivers of an endtoend communication session are not only capable of relaying and replicating data messages, but also of coding incoming messages to produce coded outgoing ones. Recent studies have shown that network coding is beneficial for peertopeer content distribution, since it eliminates the need for content reconciliation, and is highly resilient to peer failures. In this paper, we present our recent experiences with a highly optimized and highperformance C++ implementation of randomized network coding at the application layer. We present our observations based on an extensive series of experiments, draw conclusions from a wide range of scenarios, and are more cautious and less optimistic as compared to previous studies. I.
Combinatorial Optimization
, 1998
"... (All rights reserved by the authors.) C o n t e n t s ..."
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Cited by 226 (1 self)
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(All rights reserved by the authors.) C o n t e n t s
Graph minors. X. Obstructions to treedecomposition
 J. Comb. Theory, Series B
, 1991
"... Roughly, a graph has small “treewidth ” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating treewidth with the largest such obstructions (ii) a ..."
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Cited by 168 (9 self)
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Roughly, a graph has small “treewidth ” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating treewidth with the largest such obstructions (ii) an association between such obstructions and large grid minors of the graph (iii) a “treedecomposition ” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers. 0 1991 Academic Press, Inc. 1. TANGLES Graphs in this paper are finite and undirected and may have loops or multiple edges. The vertex and edgesets of a graph G are denoted by V(G) and E(G). If G, = ( V1, E,), G2 = ( V2, E2) are subgraphs of a graph G, we denote the graphs (V1n V2,E1nE,) and (V,u V2, EluEZ) by G,nG, and G, u GZ, respectively. A separation of a graph G is a pair (G,, G2) of subgraphs with G1 u G2 = G and E(G1 n G2) = 0, and the order of this separation is f V(G, n G2)(. It sometimes happens with a graph G that for each separation (G, , G2) of G of low order, we may view one of G1, G, as the “main part ” of G, in * This work ’ was performed under a consulting agreement with Bellcore.
Reverse Search for Enumeration
 Discrete Applied Mathematics
, 1993
"... The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and ..."
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Cited by 153 (25 self)
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The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing (i) all triangulations of a set of n points in the plane, (ii) all cells in a hyperplane arrangement in R d , (iii) all spanning trees of a graph, (iv) all Euclidean (noncrossing) trees spanning a set of n points in the plane, (v) all connected induced subgraphs of a graph, and (vi) all topological orderings of an acyclic graph. Finally we propose a new algorithm for the 01 integer programming problem which can be considered as an alternative to the branchandbound algorithm. 1 Introduction The listing of all objects that satisfy a specified property is a fundamental problem in combinatorics, computational geometr...
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 140 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
PseudoBoolean Optimization
 DISCRETE APPLIED MATHEMATICS
, 2001
"... This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of ..."
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Cited by 110 (4 self)
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This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of quadratic pseudoBoolean optimization (to which every pseudoBoolean optimization can be reduced), several other important special classes, and approximation algorithms.
A Functional Approach to External Graph Algorithms
 Algorithmica
, 1998
"... . We present a new approach for designing external graph algorithms and use it to design simple external algorithms for computing connected components, minimum spanning trees, bottleneck minimum spanning trees, and maximal matchings in undirected graphs and multigraphs. Our I/O bounds compete w ..."
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Cited by 90 (2 self)
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. We present a new approach for designing external graph algorithms and use it to design simple external algorithms for computing connected components, minimum spanning trees, bottleneck minimum spanning trees, and maximal matchings in undirected graphs and multigraphs. Our I/O bounds compete with those of previous approaches. Unlike previous approaches, ours is purely functionalwithout side effectsand is thus amenable to standard checkpointing and programming language optimization techniques. This is an important practical consideration for applications that may take hours to run. 1 Introduction We present a divideandconquer approach for designing external graph algorithms, i.e., algorithms on graphs that are too large to fit in main memory. Our approach is simple to describe and implement: it builds a succession of graph transformations that reduce to sorting, selection, and a recursive bucketing technique. No sophisticated data structures are needed. We apply our t...
SingleSource Unsplittable Flow
 In Proceedings of the 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The maxflow mincut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following singlesource disjoint paths problem: given a graph G, with a source vertex s and termin ..."
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Cited by 54 (2 self)
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The maxflow mincut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following singlesource disjoint paths problem: given a graph G, with a source vertex s and terminals t 1 , ..., t k , decide whether there exist edgedisjoint st i paths, for i = 1, ..., k. We consider a natural, NPhard generalization of this problem, which we call the singlesource unsplittable flow problem. We are given a source and terminals as before; but now each terminal t i has a demand ae i 1, and each edge e of G has a capacity c e 1. The problem is to decide whether one can choose a single st i path, for each i, so that the resulting set of paths respects the capacity constraints  the total amount of demand routed across any edge e must be bounded by the capacity c e . The main results of this paper are constantfactor approximation algorithms for three n...
Markov Chains and Polynomial time Algorithms
, 1994
"... This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing ..."
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Cited by 48 (0 self)
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This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.