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53
A FASTER STRONGLY POLYNOMIAL MINIMUM COST FLOW ALGORITHM
, 1991
"... In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the EdmondsKarp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n no ..."
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Cited by 116 (10 self)
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In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the EdmondsKarp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n (m + n log n)) time. Using a standard transformation, thjis approach yields an O(m log n (m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n 2 /m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say n', is much less than m. In this case, the running time of the algorithm is O((m ' + n)log n(m + n log n)).
New scaling algorithms for the assignment and minimum mean cycle problems
, 1992
"... In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing th ..."
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Cited by 50 (4 self)
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In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes, m arcs, and integer arc costs bounded by C, the algorithm runs in O(,/n m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under the similarity assumption, i.e., C = O(n k) for some k. We next consider the minimum mean cycle problem. The mean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. The minimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(~/n m log nC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.
Experimental Study of Minimum Cut Algorithms
 PROCEEDINGS OF THE EIGHTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA)
, 1997
"... Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worstcase time bounds for the problem. We conduct experimental evaluation the relative performance of these algor ..."
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Cited by 40 (2 self)
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Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worstcase time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.
Solving The Convex Cost Integer Dual Network Flow Problem
 MANAGEMENT SCIENCE
, 1999
"... In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form (i,j)Q ij ij F(w)+ iP ii B( ) ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the f ..."
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Cited by 29 (5 self)
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In this paper, we consider an integer convex optimization problem where the objective function is the sum of separable convex functions (that is, of the form (i,j)Q ij ij F(w)+ iP ii B( ) ), the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form i  j w ij , (i, j) Q), with lower and upper bounds on variables. Let n = P, m = Q, and U be the largest magnitude in the lower and upper bounds of variables. We call this problem the convex cost integer dual network flow problem. In this paper, we describe several applications of the convex cost integer dual network flow problem arising in dialaride transit problems, inverse spanning tree problem, project management, and regression analysis. We develop network flow based algorithms to solve the convex cost integer dual network flow problem. We show that using the Lagrangian relaxation technique, the convex cost integer dual network flow problem can be transformed to a convex cost primal network flow problem where each cost function is a piecewise linear convex function with integer slopes. Its special structure allows the convex cost primal network flow problem to be solved in O(nm log n log(nU)) time using a costscaling algorithm, which is the best available time bound to solve the convex cost integer dual network flow problem.
Combinatorial Algorithms for the Generalized Circulation Problem
 MATHEMATICS OF OPERATIONS RESEARCH
, 1989
"... We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e)fl(e) units arrive at the other end. For instance, nodes of the graph can correspond to different curre ..."
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Cited by 26 (3 self)
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We consider a generalization of the maximum flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e)fl(e) units arrive at the other end. For instance, nodes of the graph can correspond to different currencies, with the multipliers being the exchange rates. We require conservation of flow at every node except a given source node. The goal is to maximize the amount of flow excess at the source. This problem is a special case of linear programming, and therefore can be solved in polynomial time. In this paper we present the first polynomial time combinatorial algorithms for this problem. The algorithms are simple and intuitive.
Finding minimum cost to time ratio cycles with small integral transit times
 NETWORKS
, 1993
"... Let D = (V, E) be a digraph with n vertices and m arcs. For each e E E there is an associated cost ce and a transit time te; Ce can be arbitrary, but we require t to be a nonnegative integer. The cost to time ratio of a cycle C is X(C) = 3 ec ceCeec t. Let E ' c E denote the set of arcs e with te> ..."
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Cited by 23 (1 self)
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Let D = (V, E) be a digraph with n vertices and m arcs. For each e E E there is an associated cost ce and a transit time te; Ce can be arbitrary, but we require t to be a nonnegative integer. The cost to time ratio of a cycle C is X(C) = 3 ec ceCeec t. Let E ' c E denote the set of arcs e with te> 0, let T = max{tv: (u, v) E} for each vertex u, and let T = uev T. We give a new algorithm for finding a cycle C with the minimum cost to time ratio X(C). The algorithm's (T(m + n log n)) running time is dominated by O(T) shortest paths calculations on a digraph with nonnegative arc lengths. Further, we consider early termination of the algorithm and a faster O(Tm) algorithm in case E E ' is acyclic, i.e., in case each cycle has a strictly positive transit time, which gives an O(n²) algorithm for a class of cyclic staffing problems considered by Bartholdi et al. The algorithm can be seen to be an extension of the O(nm) algorithm of Karp for the case in which t = 1 for all e E E, which is the problem of calculating a minimum mean cycle. Our algorithm can also be modified to solve the related parametric shortest paths problem in O(T(m + n log n)) time.
Polynomial Methods For Separable Convex Optimization In Unimodular Linear Spaces With Applications
 SIAM J. Comput
, 1997
"... We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx = 0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determi ..."
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Cited by 22 (4 self)
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We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx = 0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math, 46 (1993), pp. 133165], [A. Frank and A. V. Karzanov, Technical Report RR 895M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, ...
A strongly polynomial cut canceling algorithm for minimum cost submodular flow
 PROCEEDINGS OF THE SEVENTH MPS CONFERENCE ON INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 1993
"... This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary mincost flow. The algorithm scales a relaxed optimality parameter, and creates a second, inner relaxation tha ..."
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Cited by 21 (13 self)
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This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary mincost flow. The algorithm scales a relaxed optimality parameter, and creates a second, inner relaxation that is a kind of submodular max flow problem. The outer relaxation uses a novel technique for relaxing the submodular constraints that allows our previous proof techniques to work. The algorithm uses the min cuts from the max flow subproblem as the relaxed most positive cuts it chooses to cancel. We show that this algorithm needs to cancel only O(n 3) cuts per scaling phase, where n is the number of nodes. Furthermore, we also show how to slightly modify this algorithm to get a strongly polynomial running time. Finally, we briefly show how to extend this algorithm to the separable convex cost case, and that the same technique can be used to construct a polynomial time maximum mean cut canceling algorithm for submodular flow.