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11
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 39 (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).
Use of dynamic trees in a network simplex algorithm for the maximum flow problem
, 1991
"... Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on an nvertex, marc network in at most nm pivots and O(n²m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, ..."
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Cited by 26 (4 self)
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Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on an nvertex, marc network in at most nm pivots and O(n²m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm log n). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.
Dynamic Trees as Search Trees via Euler Tours, Applied to the Network Simplex Algorithm
 Mathematical Programming
, 1997
"... The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of ..."
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Cited by 21 (1 self)
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The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of dynamic trees, values must be combined along paths. For other applications, values must be combined over entire trees. For the latter situation, we show that an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(log n) per tree operation, where n is the number of tree vertices. We apply this representation to the implementation of two versions of the network simplex algorithm, resulting in a time of O(log n) per pivot, where n is the number of vertices in the problem network.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 12 (6 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
Polynomial Dual Network Simplex Algorithms
, 1991
"... We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m² log n) bound on the number of pivots, where n and m denotes the number of nodes and a ..."
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Cited by 10 (2 self)
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We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m² log n) bound on the number of pivots, where n and m denotes the number of nodes and arcs in the input network. If the demands are integral and at most B, we also give an O(m(m+n log n) min(log nB; m log n))time implementation of a strategy that requires somewhat more pivots.
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
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We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual