Results 1  10
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36
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. COMPUT
, 1998
"... It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algori ..."
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Cited by 129 (6 self)
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It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
Compressed polytopes and statistical disclosure limitation
 ANN. INST. STATIST. MATH
, 2004
"... We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1polytope. As an application, we characterize those graphs whose cut polytopes are compressed and discuss ..."
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Cited by 32 (1 self)
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We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1polytope. As an application, we characterize those graphs whose cut polytopes are compressed and discuss consequences for studying linear programming relaxations in statistical disclosure limitation.
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 23 (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.
Improved Bounds on the MaxFlow MinCut Ratio for Multicommodity Flows
, 1993
"... In this paper we consider the worst case ratio between the capacity of mincuts and the value of maxflow for multicommodity flow problems. We improve the best known bounds for the mincut maxflow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log ..."
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Cited by 22 (2 self)
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In this paper we consider the worst case ratio between the capacity of mincuts and the value of maxflow for multicommodity flow problems. We improve the best known bounds for the mincut maxflow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log k), where D denotes the sum of all demands, and k denotes the number of commodities. In essence we prove that up to constant factors the worst mincut maxflow ratios appear in problems where demands are integral and polynomial in the number of commodities. Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the mincut maxflow ratio in general undirected graphs is bounded by O(logC log D), where C denotes the sum of all the capacities. Tragoudas has improved this bound to O(logn log D), where n is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this to O(log k log D). Klein, Plotkin and Ra...
Stronger Linear Programming Relaxations of MaxCut
 Mathematical Programming
, 2002
"... We consider linear programming relaxations for the max cut problem in graphs, based on k gonal inequalities. We show that the integrality ratio for random dense graphs is asymptotically 1 + 1=k and for random sparse graphs is at least 1 + 3=k. There are O(n ) kgonal inequalities. These results ..."
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Cited by 8 (1 self)
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We consider linear programming relaxations for the max cut problem in graphs, based on k gonal inequalities. We show that the integrality ratio for random dense graphs is asymptotically 1 + 1=k and for random sparse graphs is at least 1 + 3=k. There are O(n ) kgonal inequalities. These results generalize work by Poljak and Tuza, who gave similar results for k = 3.
Interior point and semidefinite approaches in combinatorial optimization
, 2005
"... Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient ..."
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Cited by 8 (4 self)
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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primaldual interiorpoint methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NPhard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to nonconvex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include nonconvex potential reduction methods, interior point cutting plane methods, primaldual IPMs and firstorder algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
Pfaffian graphs, tjoins, and crossing numbers
"... Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+ ..."
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Cited by 6 (0 self)
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Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1. This gives a new proof of the HananiTutte theorem. 1.
Multicommodity Flows and Approximation Algorithms
, 1994
"... This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation ..."
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Cited by 5 (0 self)
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This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation algorithm for the flux of a graph. We consider the multicommodity flow problem in which the object is to maximize the sum of the flows routed and prove the following approximate maxflow minmulticut theorem minmulticut O(log k) maxflow minmulticut where k is the number of commodities. Our proof is based on a rounding technique from [34]. Further, we show that this theorem is tight. For a multicommodity flow instance with specified demands, the ratio of the maximum concurrent flow to the sparsest cut was shown to be bounded by O(log 2 k) [30, 57, 17, 47]. We use ideas from our proof of the approximate maxflow minmulticut theorem and a geometric scaling technique from [1] to provi...
Multicommodity flows and polyhedra
 CWI QUARTERLY
, 1993
"... Seymour's conjecture on binary clutters with the socalled weak (or Q+) maxflow mincut property implies  if true  a wide variety of results in combinatorial optimization about objects ranging from matchings to (multicommodity) flows and disjoint paths. In this paper we review in particul ..."
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Cited by 4 (0 self)
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Seymour's conjecture on binary clutters with the socalled weak (or Q+) maxflow mincut property implies  if true  a wide variety of results in combinatorial optimization about objects ranging from matchings to (multicommodity) flows and disjoint paths. In this paper we review in particular the relation between classes of multicommodity flow problems for which the socalled cutcondition is sufficient and classes of polyhedra for which Seymour's conjecture is true.
A Characterization of Box 1/dintegral Binary Clutters
, 1993
"... Let Q 6 denote the port of the dual Fano matroid F 7 and let Q 7 denote the clutter consisting of the circuits of the Fano matroid F 7 that contain a given element. Let L be a binary clutter on E and let d 2 be an integer. We prove that all the vertices of the polytope fx 2 R E : x(C) 1 for ..."
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Cited by 3 (1 self)
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Let Q 6 denote the port of the dual Fano matroid F 7 and let Q 7 denote the clutter consisting of the circuits of the Fano matroid F 7 that contain a given element. Let L be a binary clutter on E and let d 2 be an integer. We prove that all the vertices of the polytope fx 2 R E : x(C) 1 for C 2 Lg " fx : a x bg are 1 d integral, for any 1 d integral a; b, if and only if L does not have Q 6 or Q 7 as a minor. Applications to graphs are presented, extending a result from [7].