Parallel primal-dual methods for the minimum cost flow problem (original) (raw)
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Efficient Parallel Algorithms for the Minimum Cost Flow Problem
Journal of Optimization Theory and Applications, 1997
In this paper, we propose efficient parallel implementations of the auction/sequential shortest path and the ∈-relaxation algorithms for solving the linear minimum cost flow problem. In the parallel auction algorithm, several augmenting paths can be found simultaneously, each of them starting from a different node with positive surplus. Convergence results of an asynchronous version of the algorithm are also given. For the ∈-relaxation method, there exist already parallel versions implemented on CM-5 and CM-2; our implementation is the first on a shared memory multiprocessor. We have obtained significant speedup values for the algorithms considered; it turns out that our implementations are effective and efficient.
Parallel Algorithms for Solving the Convex Minimum Cost Flow Problem
Computational Optimization and Applications, 2001
In this paper we deal with the solution of the separable convex cost network flow problem. In particular, we propose a parallel asynchronous version of the ∈-relaxation method and we prove theoretically its correctness. We present two implementations of the parallel method for a shared memory multiprocessor system, and we empirically analyze their numerical performance on different test problems. The preliminary numerical results show a good reduction of the execution time of the parallel algorithm with the respect to the sequential counterpart.
Imple-menting Primal-Dual Network Flow Algorithm
We show how data structures similar to those proposed recently for implementing primal simplex based codes for solving network flow problems can be used to implement primal-dual algorithms, particularly the outof-kilter algorithm. We also study several variants of a basic implementation which incorporate options for labeling, for making cost changes, for sequencing the selection of out-of-kilter arcs, and for implementing the primal-dual algorithm. Our investigations indicate that storing and manipulating data efficient leads to substantial reductions in computation time as well as storage requirements.
Parallel Computing Solution for Capacity Expansion Network Flow Optimization Problems
In this work, an algorithm is proposed to transform a specific capacity expansion network flow problem to a classical linear network flow problem so that it can be solved with high performance. In the specific network flow problem, the total flow demand is greater than the total flow supply and each node needs to create flow to satisfy its own flow demand and/or other nodes' flow demand such that the total flow demand is satisfied and the total flow generation and transmission capacity is minimized. This provides a very efficient solution for some real-world network problems such as commodity flow planning and scheduling problems, power system capacity expansion planning problems, data communication network problems, transportation and telecommunication network capacity expansion planning problems, airline scheduling problems, and circulation–demand problems. In classical linear network flow (LNF) problems, a network consists of multiple source and sink nodes, where each node is a sink node or a source node, but not both. Usually, there is only one kind of commodity flow and the goal is to find flow schedules and routes such that all sink nodes’ flow demands are satisfied and the total flow transmission cost is minimized. We develop a capacity expansion multicommodity network flow (CEMNF) problem, in which the total commodity supply is less than the total commodity demand. There are more than one kind of commodities and each node is a commodity flow generator, as well as a consumer. It is allowed to do expansion for commodity flow generation capacities at each node and also to do expansion for commodity flow capacities of each arc so that more flow can be transmitted among nodes. Thus, CEMNF is not only a commodity flow routing problem, but also a commodity generation and flow planning problem, in which the increasing commodity demands need to be satisfied by generation/transmission capacity expansions. The goal of CEMNF problems is to find the flow routes and capacity expansion plans such that all flow demands are satisfied and the total cost of routing and planning is minimized. High-performance distributed computing algorithms have been designed to solve classical linear network flow (LNF) problems have been proposed. Solving the general CEMNF problems by high-performance distributed computing algorithms is an open research question. The LNF problems can be formulated as linear programming models and algorithms have been proposed to solve them efficiently on distributed computing platforms. But, the constraints of the CEMNF problems do not allow them to solve using the same methodology. In this paper, we also develop a transformation method to transform CEMNF problems into LNF problems in polynomial time and space complexity to solve them efficiently on distributed computing platforms. The results show that we can solve CEMNF problems with high performance.
Polynomial-time primal simplex algorithms for the minimum cost network flow problem
Algorithmica, 1992
We present two variants of the primal network simplex algorithm which solve the minimum cost network flow problem in at most O(n2m 2 log n) pivots. Here we define the network simplex method as a method which proceeds from basis tree to adjacent basis tree regardless of the change in objective function value; i.e., the objective function is allowed to increase on some iterations. The first method is an extension of the minimum mean augmenting cycle-canceling method of Goldberg and Tarjan. The second method is a combination of a cost-scaling technique and a primal network simplex method for the maximum flow problem. We also show that the diameter of the primal network flow polytope is at most n2m.
Massively Parallel Solution of Large Scale Network Flow Problems
Nonlinear Optimization and Applications, 1996
Two massively parallel algorithms for large scale linear and convex quadratic network ow problems are proposed and studied. The methods are based on the alternating step method for monotropic programming. The original network ow problem is decomposed in simple subproblems involving only few variables for which solution in closed form exists. Computational results obtained on the CRAY T3D show that the methods hold the promise of solving extremely large size problems.
A Primal Algorithm for Finding Minimum-Cost Flows in Capacitated Networks With Applications
Bell System Technical Journal, 1982
Algorithms for finding a minimum-cost, single-commodity flow in a capacitated network are based on variants of the simplex method of linear programming. We describe an implementation of a primal algorithm which is fast and can solve large problems. The major ideas incorporated are (i) the sparsity ofthe network is used to reduce the time and computer storage space requirements; (ii) basic solutions are stored compactly as spanning trees of the network; (iii) a candidate stack is used to allow flexible strategies in choosing an arc to enter the basis tree; (iv) the predecessor and thread data structures are used to efficiently traverse the tree and to update the solution at each iteration; (v) rules are implemented to avoid cycling or stalling caused by degeneracy; and (vi)piecewise-linear, convex arc costs are handled implicitly. The Primal Network Flow Convex (PNFC) code implements this algorithm and three examples, from communication networks, that can be solved with PNFC are discussed: (i) solving the area transfer problem; (ii) scheduling the collection of traffic data records; and (iii) planning the placement ofpair-gain systems.
Parallel Computing, 1997
In this paper we present a parallel asynchronous implementation of the e-relaxation method for solving the linear minimum cost flow problem on distributed memory message-passing multiprocessor systems. The general structure of the method is well suited to efficient parallelization, since a single iteration can be performed on several nodes simultaneously. We describe the implementation details of the parallel version on both a Fujitsu APlOOO and a cluster of Digital Alpha workstations connected via FDDI links. The results obtained demonstrate that our implementation is capable of substantial speedups.
Partially asynchronous, parallel algorithms for network flow and other problems.
SIAM J. CONTROL OPTIMIZ., 1990
The problem of computing a fixed point of a nonexpansive function f is considered. Sufficient conditions are provided under which a parallel, partially asynchronous implementation of the iteration x:=f(x) converges. These results are then applied to (i) quadratic programming subject to box constraints, (ii) strictly convex cost network flow optimization, (iii) an agreement and a Markov chain problem, (iv) neural network optimization, and (v) finding the least element of a polyhedral set determined by a weakly diagonally dominant, Leontief system. Finally, simulation results illustrating the attainable speedup and the effects of asynchronism are presented.
Dual Based Procedure for the Single Stage General Minimum Cost Flow Problem
2015
Abstract— In this paper, we consider a single stage un-capacitated minimum cost flow network and device two novel procedures to obtain very good dual and primal solutions. We demonstrate the working of heuristics on two sample problems. Later we extend the ideas developed to a general min cost flow network problem by solving a representative problem. We further plan to specialize this procedure for single stage capacitated minimum cost flow network problem and general capacitated minimum cost flow problem.