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33
Network Coding: The Case of Multiple Unicast Sessions
 in Proceedings of the 42nd Allerton Annual Conference on Communication, Control, and Computing
, 2004
"... In this paper, we investigate the benefit of network coding over routing for multiple independent unicast transmissions. We compare the maximum achievable throughput with network coding and that with routing only. We show that the result depends crucially on the network model. In directed network ..."
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Cited by 54 (7 self)
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In this paper, we investigate the benefit of network coding over routing for multiple independent unicast transmissions. We compare the maximum achievable throughput with network coding and that with routing only. We show that the result depends crucially on the network model. In directed networks, or in undirected networks with integral routing requirement, network coding may outperform routing. In undirected networks with fractional routing, we show that the potential for network coding to increase achievable throughput is equivalent to the potential of network coding to increase bandwidth e#ciency, both of which we conjecture to be nonexistent.
A Polyhedral Approach to Multicommodity Survivable Network Design
 Numerische Mathematik
, 1993
"... The design of costefficient networks satisfying certain survivability ..."
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Cited by 32 (0 self)
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The design of costefficient networks satisfying certain survivability
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
Costefficient network synthesis from leased lines
, 1997
"... Given a communication demand between each pair of nodes of a network we consider the problem of deciding what capacity to install on each edge of the network in order to minimize the building cost of the network and to satisfy the demand between each pair of nodes. The feasible capacities that can b ..."
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Cited by 16 (2 self)
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Given a communication demand between each pair of nodes of a network we consider the problem of deciding what capacity to install on each edge of the network in order to minimize the building cost of the network and to satisfy the demand between each pair of nodes. The feasible capacities that can be leased from a network provider are of a particular kind in our case. There are a few socalled basic capacities having the property that every basic capacity is an integral multiple of every smaller basic capacity. An edge can be equipped with a capacity only if it is an integer combination of the basic capacities. We treat, in addition, several restrictions on the routings of the demands (length restriction, diversification) and failures of single nodes or single edges. We formulate the problem as a mixed integer linear programming problem and develop a cutting plane algorithm as well as several heuristics to solve it. We report on computational results for real world data.
On skeletons, diameters and volumes of metric polyhedra
 Combinatorics and Computer Science, Lecture
"... Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency a ..."
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Cited by 15 (10 self)
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Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the maxcut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm:ons and connectivity of the metric polytope and its relatives. In partic~dar, using its large symmetry group, we completely describe all the 13 o:bits which form the 275 840 vertices of the 21dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the/skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method. 1
On the Skeleton of the Metric Polytope
, 2001
"... We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known instan ..."
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Cited by 10 (1 self)
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We consider convex polyhedra with applications to wellknown combinatorial optimization problems: the metric polytope mn and its relatives. For n # 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7  the largest previously known instance  has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 vertices of the 28dimensional metric polytope m8 . The description consists of 533 orbits and is conjectured to be complete. The orbitwise incidence and adjacency relations are also given. The skeleton of m8 could be large enough to reveal some general features of the metric polytope on n nodes. While the extreme connectivity of the cuts appears to be one of the main features of the skeleton of mn , we conjecture that the cut vertices do not form a cutset. The combinatorial and computational applications of this conjecture are studied. In particular, a heuristic skipping the highest degeneracy is presented. 1
On the Solitaire Cone and Its Relationship to MultiCommodity Flows
 PREPRINT CAMS 142 ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES
, 2001
"... The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities ov ..."
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Cited by 7 (3 self)
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The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NPcompleteness result; 3. a method of generating large classes of facets; 4. a complete characterization of 01 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards; 7. a complete characterization of the adjacency of extreme rays, diameter, number of 2faces and edge connectivity for rectangular toric boards.
Computational Experience with the Reverse Search Vertex Enumeration Algorithm
 Optimization Methods and Software
, 1998
"... Dedicated to Professor Masao Iri on the occasion of his 65th birthday This paper describes computational experience obtained in the development of the lrs code, which uses the reverse search technique to solve the vertex enumeration/convex hull problem for ddimensional convex polyhedra. We giv e em ..."
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Cited by 7 (2 self)
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Dedicated to Professor Masao Iri on the occasion of his 65th birthday This paper describes computational experience obtained in the development of the lrs code, which uses the reverse search technique to solve the vertex enumeration/convex hull problem for ddimensional convex polyhedra. We giv e empirical results showing improvements obtained by the use of lexicographic perturbation, lifting, and integer pivoting. We also give some indication of the cost of using extended precision arithmetic and illustrate the use of the estimation function of lrs. The empirical results are obtained by running various versions of the program on a set of wellknown nontrivial polyhedra: cut, configuration, cyclic, Kuhn_Quandt, and metric polytopes. Ke ywords: vertex enumeration, convex hulls, reverse search, computational experience 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.
Tight spans of distances and the dual fractionality of undirected multiflow problems
, 2009
"... In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Karzanov. The first one is a characterization of commodity graphs H for which the dual ..."
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Cited by 6 (6 self)
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In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Karzanov. The first one is a characterization of commodity graphs H for which the dual of maximum multiflow problem with respect to H has bounded fractionality, and the second one is a characterization of metrics d on terminals for which the dual of metricweighed maximum multiflow problem has bounded fractionality. A key ingredient of the present paper is a nonmetric generalization of the tight span, which was originally introduced for metrics by Isbell and Dress. A theory of nonmetric tight spans provides a unified duality framework to the weighted maximum multiflow problems, and gives a unified interpretation of combinatorial dual solutions of several known minmax theorems in the multiflow theory.