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34
Faster scaling algorithms for network problems
 SIAM J. COMPUT
, 1989
"... This paper presents algorithms for the assignment problem, the transportation problem, and the minimumcost flow problem of operations research. The algorithms find a minimumcost solution, yet run in time close to the bestknown bounds for the corresponding problems without costs. For example, the ..."
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Cited by 163 (5 self)
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This paper presents algorithms for the assignment problem, the transportation problem, and the minimumcost flow problem of operations research. The algorithms find a minimumcost solution, yet run in time close to the bestknown bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimumcost matching in a bipartite graph) can be solved in O(v/’rn log(nN)) time, where n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.
Fast and Robust Earth Mover’s Distances
"... We present a new algorithm for a robust family of Earth Mover’s Distances EMDs with thresholded ground distances. The algorithm transforms the flownetwork of the EMD so that the number of edges is reduced by an order of magnitude. As a result, we compute the EMD by an order of magnitude faster tha ..."
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Cited by 90 (6 self)
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We present a new algorithm for a robust family of Earth Mover’s Distances EMDs with thresholded ground distances. The algorithm transforms the flownetwork of the EMD so that the number of edges is reduced by an order of magnitude. As a result, we compute the EMD by an order of magnitude faster than the original algorithm, which makes it possible to compute the EMD on large histograms and databases. In addition, we show that EMDs with thresholded ground distances have many desirable properties. First, they correspond to the way humans perceive distances. Second, they are robust to outlier noise and quantization effects. Third, they are metrics. Finally, experimental results on image retrieval show that thresholding the ground distance of the EMD improves both accuracy and speed. 1.
Improved Time Bounds for the Maximum Flow Problem
, 1987
"... A Recently, Goldberg proposed a new approach to the maximum network flow problem. The approach yields a very simple algorithm running in O(n) "time on nvertex networks. Incorporation of the dynamic tree data structure of Sleator and Tarjan yields a more complicated algorithm with a running ti ..."
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Cited by 48 (11 self)
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A Recently, Goldberg proposed a new approach to the maximum network flow problem. The approach yields a very simple algorithm running in O(n) "time on nvertex networks. Incorporation of the dynamic tree data structure of Sleator and Tarjan yields a more complicated algorithm with a running time of O(nm log (n2/m)) on medge netvorks.'Ahuja and Orlin developedIa variant of GoldbeWrs algorithm, that uses scaling and runs in O(nm + h2 logU) time on networks with integer edge capacities bounded by U. * this paper w. t obtaina modification of the AhujaOrlin algorithm with a running time of 0(nm + n2 loU). We4heshow thatthe use of dynamic trees in this algologlogUrithm further reduces the time bound to 0(nm log ( n logU + 2)). This result m loglogU demonstrates that the combined use of scaling and dynamic trees results in speed
Improved approximation algorithms for unsplittable flow problems (Extended Abstract)
 IN PROCEEDINGS OF THE 38TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1997
"... In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by it ..."
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Cited by 44 (2 self)
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In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by its capacity c e : The problem is an NPhard variant of max flow and a generalization of singlesource edgedisjoint paths with applications to scheduling, load balancing and virtualcircuit routing problems. In a significant development, Kleinberg gave recently constantfactor approximation algorithms for several natural optimization versions of the problem [18]. In this paper we give a generic framework that yields simpler algorithms and significant improvements upon the constant factors. Our framework, with appropriate subroutines, applies to all optimization versions previously considered and treats in a unified manner directed and u...
A polynomial time primal network simplex algorithm for minimum cost flows
, 1995
"... Developing a polynomial time algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2 m log nC, n 2 m 2 log n)) time, where n is the number of nodes in the network, m is the number of arcs, and C denotes th ..."
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Cited by 37 (1 self)
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Developing a polynomial time algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2 m log nC, n 2 m 2 log n)) time, where n is the number of nodes in the network, m is the number of arcs, and C denotes the maximum absolute arc costs if arc costs are integer and 0 otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the &quot;premultiplier algorithm. &quot; A vector X of node potentials is called a vector of premultipliers with respect to a rooted tree if each arc directed towards the root has a nonpositive reduced cost and each arc directed away from the root has a nonnegative reduced cost. We then develop a costscaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm log nC, nm 2 log n)) pivots, With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm log n).
Algorithms for dense graphs and networks on the random access computer
 ALGORITHMICA
, 1996
"... We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to com ..."
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Cited by 28 (4 self)
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We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to compute the transitive closure of a digraph with n vertices and m edges in time O(n 2 + nm/,k), how to solve the uncapacitated transportation problem with integer costs in the range [0..C] and integer demands in the range [U..U] in time O ((n 3 (log log / log n) 1/2 + n 2 log U) log nC), and how to solve the assignment problem with integer costs in the range [0..C] in time O(n 2"5 log nC/(logn/loglog n)l/4). Assuming a suitably compressed input, we also show how to do depthfirst and breadthfirst search and how to compute strongly connected components and biconnected components in time O(n~. + n2/L), and how to solve the single source shortestpath problem with integer costs in the range [0..C] in time O(n²(log C)/log n). For the transitive closure algorithm we also report on the experiences with an implementation.
Approximation algorithms for singlesource unsplittable flow
 SIAM Journal on Computing
, 2002
"... In the singlesource unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain sinks in a given graph with edge capacities. The demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacit ..."
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Cited by 25 (4 self)
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In the singlesource unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain sinks in a given graph with edge capacities. The demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity. This problem was introduced by Kleinberg [1996a] and generalizes several NPcomplete problems. A cost value per unit of flow may also be defined for every edge. In this paper, we implement the 2approximation algorithm of Dinitz, Garg, and Goemans [1999] for congestion, which is the best known, and the (3, 1)approximation algorithm of Skutella [2002] for congestion and cost, which is the best known bicriteria approximation. We study experimentally the quality of approximation achieved by the algorithms and the effect of heuristics on their performance. We also compare these algorithms against the previous best ones by Kolliopoulos and Stein [1999] Categories and Subject Descriptors: G.2.2 [Discrete Mathematics]: Graph Algorithms—Graph
Belief Propagation for Mincost Network Flow: Convergence & Correctness
"... We formulate a Belief Propagation (BP) algorithm in the context of the capacitated minimumcost network flow problem (MCF). Unlike most of the instances of BP studied in the past, the messages of BP in the context of this problem are piecewiselinear functions. We prove that BP converges to the opti ..."
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Cited by 17 (2 self)
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We formulate a Belief Propagation (BP) algorithm in the context of the capacitated minimumcost network flow problem (MCF). Unlike most of the instances of BP studied in the past, the messages of BP in the context of this problem are piecewiselinear functions. We prove that BP converges to the optimal solution in pseudopolynomial time, provided that the optimal solution is unique and the problem input is integral. Moreover, we present a simple modification of the BP algorithm which gives a fully polynomialtime randomized approximation scheme (FPRAS) for MCF. Thisisthe first instance where BP is proved to have fullypolynomial running time. 1