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We propose an algorithm for constructing orthogonal n-ary operations which is called a block composition algorithm here. Input data of the algorithm are two series of different arity operations being distributed by blocks. The algorithm consists of two parts: composition algorithm for constructing n-ary operations with orthogonal retracts from given blocks of operations and block-wise recursive algorithm for constructing orthogonal n-ary operations from obtained operations. Obtained results are illustrated by examples of orthogonal n-ary operations which are constructible by block-wise recursive algorithm and non-constructible by the well-known trivial recursive algorithm.

This paper proposes an efficient segregated solution procedure for the two-dimensional incompressible fluid flow on unstructured grids. The new algorithm is called SIMPLERR (SIMPLERR-revised algorithm). It includes an inner iterative process that consists of a pressure equation solution and an explicit velocity correction, and then, the pressure field is obtained by another solving process for the pressure equation. The features and advantages of the SIMPLERR algorithm are demonstrated by solving a benchmark flow problem, and the results indicate that the SIMPLERR algorithm can maintain a strong stability as the IDEAL algorithm and can converge faster than the SIMPLER algorithm or even than the IDEAL algorithm. The advantage of the SIMPLERR algorithm is especially evident for high-Re and fine-mesh fluid flow cases, where the SIMPLERR algorithm and the IDEAL algorithm can obtain a convergence result but the SIMPLER algorithm cannot. The SIMPLERR algorithm is about two times faster than the IDEAL al...

Flow variations over time generalize standard network flows by introducing an element of time. In contrast to the classical case of static flows, a flow over time in such a network specifies a flow rate entering an arc for each point in time. In this setting, the capacity of an arc limits the rate of flow into the arc at each point in time. Traditionally, flows over time are computed in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static network flows, its drawback is the enormous size of the time-expanded network. In this paper, we extend the results about the minimum flow problem to network flows (with n nodes and m arcs) in which the time-varying lower bounds can involve both the source and the sink nodes (as in Fathabadi et al.) and also one additional node other than the source and the sink nodes. It is shown that this problem for the set \(\{0,1,\ldots ,T\}\) of time points can be solved by at most n minimum flow computations, by suitably extending the dynamic minimum flow algorithm and reoptimization techniques. The running time of the presented algorithm is \(O(n^2m)\).

Given a directed graph with a capacity and a transit time for each arc and with single source and single sink nodes, the quickest flow problem is to find the minimum time horizon to send a given amount of flow from the source to the sink. This is one of the fundamental dynamic flow problems. Parametric search is one of the basic approaches to solving the problem. Recently, Lin and Jaillet (SODA, 2015) proposed an algorithm whose time complexity is the same as that of the minimum cost flow algorithm. Their algorithm employs a cost scaling technique, and its time complexity is weakly polynomial time. In this article, we modify their algorithm by adopting a technique to construct a strongly polynomial time algorithm for solving the minimum cost flow problem. The proposed algorithm runs in O ( n m 2 ( log n ) 2 ) time, where n and m are the numbers of nodes and arcs, respectively. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

We investigate variants of the max-flow problem in a network under k attacks. The network interdiction problem is to find the minimum max-flow value among (mk) networks that can be obtained by deleting each set of k links. The adaptive network flow problem is to find a flow of the network such that the flow value is maximum against any set of k links attack, when deleting the corresponding flow to those k links in the original flow. First, we prove that max-(k+1)-route flow is a (k+1)-approximation for both problems. Also, we develop a polynomial-time heuristic algorithm for both cases, called the iterative multiroute flow. Then in a second phase, we investigate properties of the function taking the real value h to the max h-route flow value, and apply the result to solve both of the problems. We show that the function is piecewise hyperbolic, and modify a standard parametric optimization technique to find this function. The running time of the algorithm is O(T), when is a sourcesink edge connectivity of our network and T the computation time of a max-flow algorithm. We show that for some instances, when h is optimally chosen, the max- h-route flow is an exact solution for both problems. Maximum multiroute flow algorithm is a (k+1)-approximation algorithm for the network interdiction and adaptive network flow problems.If a parameter of the algorithm is optimally adjusted, the algorithm can be an exact algorithm when the input network satisfies a condition.Experimental results shown that most of the problem instances satisfy the condition.

Maximum Flow Problem (MFP) discusses the maximum amount of flow that can be sent from the source to sink. Edmonds-Karp algorithm is the modified version of Ford-Fulkerson algorithm to solve the MFP. This paper presents some modifications of Edmonds-Karp algorithm for solving MFP. Solution of MFP has also been illustrated by using the proposed algorithm to justify the usefulness of proposed method.

We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities. Our algorithm views the allocation of money as flows and iteratively improves the balanced flow as in 11] for Fisher's model. We develop new methods to carefully deal with the flows and surpluses during price adjustments. Our algorithm performs O ( n 6 log ? ( n U ) ) maximum flow computations, where n is the number of agents and U is the maximum integer utility. The flows have to be presented as numbers of bitlength O ( n log ? ( n U ) ) to guarantee an exact solution. Previously, 22,29] have given polynomial time algorithms for this problem, which are based on solving convex programs using the ellipsoid algorithm and the interior-point method, respectively.

We consider the problem of finding a maximum flow in networks with strictly and nonstrictly determined flow distribution. We show that, for each condition of flow distribution, a solution to the problem under consideration exists and is unique. Some algorithms for finding a maximum flow are developed for each condition of flow distribution. We find bounds for the maximum flow value in networks with strict flow distribution.

Most of the techniques of nonlinear optimization assume the existence of a differential function, but there also exist other approaches. In this paper, we propose a non-differential algorithm that solves the problem of minimization and maximization for any objective function of several unknown variables. The algorithm searches for a solution vector of variables that yields the optimal result of the objective function, using a trial-and-error process. The algorithm is not fast, but it converges to a non-trivial solution, if it exists. The method will be used to determine the minimum and the maximum of the outputs of a socio-political model of E-Democracy. This model of E-Democracy is based on a Mamdani fuzzy inference system. We provide formalization and Matlab implementation for our fuzzy system and for the algorithm

This paper investigates uncertain maximum flow problem. To obtain the uncertainty distribution of the maximum flow, an α -maximum flow model is formulated under uncertainty theory framework. It is proved that there exists an equivalence relationship between the α -maximum flow model and the classic deterministic maximum flow model. Furthermore, some important properties of α -maximum flow model are analyzed, based on which a polynomial algorithm is proposed. Finally, a numerical example is presented to illustrate the model and the algorithm.

The paper presents a 1.692nnO(1)-time polynomial-space algorithm for the traveling salesman problem in an n-vertex edge-weighted graph with maximum degree 4, which improves the previous results of the 1.890nnO(1)-time polynomial-space algorithm by Eppstein and the 1.733nnO(1)-time exponential-space algorithm by Gebauer.

The Map Maker algorithm which converts survey data into geometric data with 2-dimensional Cartesian coordinates has been previously published. Analysis of the performance of this algorithm is continuing. The algorithm is suitable for generating 2D maps and it would be helpful to have this algorithm generalized to generate 3D and higher dimensional coordinates. The trigonometric approach of the Map Maker algorithm does not extend well into higher dimensions however this paper reports on an algebraic approach which solves the problem. A similar algorithm called the Coordinatizator algorithm has been published which converts survey data defining a higher dimensional space of measured sites into the lowest dimensionalcoordinatization accurately fitting the data. Therefore the Coordinatizator algorithm is not a projection transformation whereas the n-dimensional Map Maker algorithm is.

A new Evolutionary algorithm was presented based on intermediate value theorem of continuous function. The global search ability and the local search ability of this algorithm are well balanced, and operators used in the algorithm are simple, Further, small size of population scale is used. Initial numerical experiments show that the mean evolutionary algorithm is better than differential evolution algorithm in solving high dimension function optimization.

The optimization of the Electromagnetism-like Mechanism (EM) algorithm is analyzed, and a Hybrid Chaos Search (CS) Electromagnetism-like Mechanism Algorithm (CEM) is proposed in this paper. The new algorithm combines the advantages of both the CS method and the EM algorithm. To make the initial population more uniform, the CEM algorithm constructs it with the theory of good point set in number theory. The CS method, rather than the random linear search algorithm, is used in the new algorithm, which can effectively prevent the search process from terminating prematurely. Besides, an adaptive movement is used in the improved algorithm to speed up convergence, and a genetic coefficient is added to the update formula to update the locations of particles so that the particles are more likely to move into the other feasible regions. Experimental results show that the improved algorithm can converge to global optimums more effectively and accurately.

In this paper, we are concerned with maximum flow problems with non-zero lower bounds. The common approach to this problem is transforming the network into a bigger one with zero lower bounds, whose optimal solution yields a feasible solution to the original problem and then using one of the established methods for maximum flow problems with little to no modifications. Expanding upon the labelling techniques of Goldfarb and Hao we show that a variant of the monotonic build-up simplex algorithm runs in strongly polynomial time on the original network. The main characteristic of the MBU algorithm is that starting from an arbitrary basis solution it decreases the number of infeasible variables monotonically, without letting any feasible variables turn infeasible in the process. We show that this algorithm terminates after at most pivots which makes it the first strongly polynomial pivot algorithm that solves the problem without transforming the network.

This paper presents a heuristic algorithm for the earliest arrival flow problem. Existing exact algorithms, even polynomial in the output size, contain submodular function optimization as a frequently called subroutine, and thus are not practical in real-life applications. In this paper we propose an algorithm that does not involve the submodular function optimization. Although solving an EAF near-optimal, the algorithm is remarkably simple and efficient as it only involves shortest path computations on a static network. A numerical example illustrates how the algorithm works. As an application, we demonstrate the algorithm’s solution quality and computational performance by solving a real-size network.

The ill-condition of the load-flow is caused by a variety of reasons. Among them, heavy load and a long line of radiation type slip(large r/x) are the two important reasons. In connection with the ill-conditon above, Newton-homotopy algorithm is introduced in this paper. The algorithm mixes the locally quadratic convergence of Newton-raphison with wide range convergence of homopoty algorithm, so it improves the ill-condition of Jacobian matrix in the process of load-flow calculating, and speeds the convergence. Finally, several examples show the efficiency of the algorithm.

The capacity scaling algorithm for the maximum flow problem runs in O(nmlogU) time where n is the number of nodes, m is the number of arcs, and U is the largest arc capacity in the network. The two-phase capacity scaling algorithm reduces this bound to O(nmlog(U/n)). This running time is achieved with the restriction that flows are pushed on individual arcs while paths are being identified, but this causes slower empirical run times compared to the single-phase capacity scaling algorithm. In this research, we prove that the two-phase algorithm runs in the same theoretical time without the mentioned restriction. We also show that in practice, it runs significantly faster than the single-phase capacity scaling algorithm.

The constrained maximum flow problem is a variant of the classical maximum flow problem in which the flow from a source node to a sink node is maximized in a directed capacitated network with arc costs subject to the constraint that the total cost of flow should be within a budget. It is important to study this problem because it has important applications, such as in logistics, telecommunications and computer networks; and because it is related to variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications as well. In this research, we present an O(n^2mlog(nC)) time cost scaling algorithm and compare its empirical performance against the two existing polynomial combinatorial algorithms for the problem: the capacity scaling and the double scaling algorithms. We show that the cost scaling algorithm is on average 25 times faster than the double scaling algorithm, and 32 times faster than the capacity scaling algorithm.

The routing problems pertain to the search for a shortest route (minimum cost or minimum distance or maximum flow, etc.) connecting two specified stations or nodes described as ‘source’ and ‘sink’. The paper aims at obtaining an optimal route of a more realistic situation as to scheduling maximum flows at a minimum cost from a source to a destination. The distance (cost) and arc capacity between any two stations are given. The objective is to find the maximum flow with the minimum cost from the source to the destination in a network. The problem has many applications in the field of network flow theories. Several special cases of the problem were intensively studied in the literature and proposed various techniques to solve. Here we solved the present problem by the lexicographic search technique, which gives the exact solution to the problem. The solution procedure is illustrated with a suitable example. The algorithm is also tested in C-language and the computational details are also reported.

To solve the static traffic grooming problems for differentiated traffic model a MO — DSG algorithm is proposed. The algorithm defines the business value to describe the requirements for QoS performance. Let traffic grooming method be equivalent to gene, and grooming scheme to chromosome, hence the original problem of traffic grooming is transformed into a typical mathematical model which can be solved by multi target optimization algorithm. To prove the superiority of the algorithm, we compared the proposed algorithm with the NSGA-II algorithm by using the classical test functions. In addition, simulation based on the actual scene of MO-DSG algorithm proves its superior performance to TBSG-c algorithm, IONT algorithm, on construction cost and risk.

The Vortex Search (VS) algorithm is one of the recently proposed metaheuristic algorithms which was inspired from the vortical flow of the stirred fluids. Although the VS algorithm is shown to be a good candidate for the solution of certain optimization problems, it also has some drawbacks. In the VS algorithm, candidate solutions are generated around the current best solution by using a Gaussian distribution at each iteration pass. This provides simplicity to the algorithm but it also leads to some problems along. Especially, for the functions those have a number of local minimum points, to select a single point to generate candidate solutions leads the algorithm to being trapped into a local minimum point. Due to the adaptive step-size adjustment scheme used in the VS algorithm, the locality of the created candidate solutions is increased at each iteration pass. Therefore, if the algorithm cannot escape a local point as quickly as possible, it becomes much more difficult for the algorithm to escape from that point in the latter iterations. In this study, a modified Vortex Search algorithm (MVS) is proposed to overcome above mentioned drawback of the existing VS algorithm. In the MVS algorithm, the candidate solutions are generated around a number of points at each iteration pass. Computational results showed that with the help of this modification the global search ability of the existing VS algorithm is improved and the MVS algorithm outperformed the existing VS algorithm, PSO2011 and ABC algorithms for the benchmark numerical function set.

Based on rough graph theory, this paper gives a new algorithm in rough network, which generalizes the classical algorithm of exploring maximum flow. This algorithm successfully deals with a kind of complex relationship mining problem between different relationship levels. Simulation shows the effectiveness of this algorithm.

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O ( mn log ( nU )), where U is the value of maximum upper bound. Our algorithm improves the O ( m 2 log ( nU ))-time algorithm in [12] . Hence the current algorithm improves the running time in [12] by a factor of n . Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O ( mn log n ) time, see [5] . Under similarity assumption [11] , our algorithm runs in O ( mn log n ) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.

The well-known sequential minimal optimization (SMO) algorithm is the most commonly used algorithm for numerical solutions of the support vector learning problems. At each iteration in the traditional SMO algorithm, also called 2PSMO algorithm in this paper, it jointly optimizes only two chosen parameters. The two parameters are selected either heuristically or randomly, whilst the optimization with respect to the two chosen parameters is performed analytically. The 2PSMO algorithm is naturally generalized to the three-parameter sequential minimal optimization (3PSMO) algorithm in this paper. At each iteration of this new algorithm, it jointly optimizes three chosen parameters. As in 2PSMO algorithm, the three parameters are selected either heuristically or randomly, whilst the optimization with respect to the three chosen parameters is performed analytically. Consequently, the main difference between these two algorithms is that the optimization is performed at each iteration of the 2PSMO algorithm on a line segment, whilst that of the 3PSMO algorithm on a two-dimensional region consisting of infinitely many line segments. This implies that the maximum can be attained more efficiently by 3PSMO algorithm. Main updating formulae of both algorithms for each support vector learning problem are presented. To assess the efficiency of the 3PSMO algorithm compared with the 2PSMO algorithm, 14 benchmark datasets, 7 for classification and 7 for regression, will be tested and numerical performances are compared. Simulation results demonstrate that the 3PSMO outperforms the 2PSMO algorithm significantly in both executing time and computation complexity.

Maximum-flow problems occur in a wide range of applications. Although already well-studied, they are still an area of active research. The fastest available implementations for determining maximum flows in graphs are either based on augmenting-path or on push-relabel algorithms. In this work, we present two ingredients that, appropriately used, can considerably speed up these methods. On the theoretical side, we present flow-conserving conditions under which subgraphs can be contracted to a single vertex. These rules are in the same spirit as presented by Padberg and Rinaldi (Math. Programming (47), 1990) for the minimum cut problem in graphs. These rules allow the reduction of known worst-case instances for different maximum flow algorithms to equivalent trivial instances. On the practical side, we propose a two-step max-flow algorithm for solving the problem on instances coming from physics and computer vision. In the two-step algorithm flow is first sent along augmenting paths of restricted lengths only. Starting from this flow, the problem is then solved to optimality using some known max-flow methods. By extensive experiments on instances coming from applications in theoretical physics and in computer vision, we show that a suitable combination of the proposed techniques speeds up traditionally used methods.

The minimum cut problem is to minimize c(S,T), that is, to determine source S and sink T such that the capacity of the S-T cut is minimal. The flow-based algorithm is mostly used to find the bottleneck arcs by calculating flow network, and does not presents the minimum cut. This paper suggests an algorithm that simply includes the maximum capacity vertex to adjacent set S or T and finds the minimum cut without obtaining flow network in advance. On applying the suggested algorithm to 13 limited graphs, it can be finds the minimum cut value (S, T) with simply and correctly.

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.

This paper considers a new class of network flows, called dynamic generative network flows in which, the flow commodity is dynamically generated at a source node and dynamically consumed at a sink node and the arc-flow bounds are time dependent. Then the maximum dynamic flow problem in such networks for a pre-specified time horizon T is defined and mathematically formulated in both arc flow and path flow presentations. By exploiting the special structure of the problem, an efficient algorithm is developed to solve the general form of the dynamic problem as a minimum cost static flow problem.

The Minimum Capacity s-t Cut Problem (MinCut) is an intensively studied problem in combinatorial optimization. A natural extension is the problem of choosing a minimum capacity s-t cut when arc capacities are unknown but confined to known intervals. This motivates the Robust Minimum Capacity s-t Cut Problem (RobuCut), which has applications such as open-pit mining and project scheduling. In this paper, we show how RobuCut can be reduced to solving a sequence of maximum flow problems and provide an efficient algorithm for rapidly solving this sequence of problems. We demonstrate that our algorithm solves instances of RobuCut in seconds that would require hours if a standard maximum flow solver is iteratively used as a black-box subroutine.

In this paper, we propose a fast heuristic algorithm for the maximum concurrent k-splittable flow problem. In such an optimization problem, one is concerned with maximizing the routable demand fraction across a capacitated network, given a set of commodities and a constant k expressing the number of paths that can be used at most to route flows for each commodity. Starting from known results on the k-splittable flow problem, we design an algorithm based on a multistart randomized scheme which exploits an adapted extension of the augmenting path algorithm to produce starting solutions for our problem, which are then enhanced by means of an iterative improvement routine. The proposed algorithm has been tested on several sets of instances, and the results of an extensive experimental analysis are provided in association with a comparison to the results obtained by a different heuristic approach and an exact algorithm based on branch and bound rules.

This study describes two types of algorithms for skid-to-turn unmanned aerial vehicles to plan paths between two waypoints under constant wind conditions. The first type of algorithm is a rigorous optimization algorithm based on the Euler-Lagrange formulation with analytical integration of the path. The second type of algorithm is a fast algorithm describing the path by two circular arcs connected by a line segment or another circular arc in the air mass frame, which is similar to the Dubins path. The latter algorithm is developed for actual airborne application, whereas the former algorithm is developed to check the quasi-optimality of the path calculated by the latter algorithm. We present a convergence proof of the latter algorithm under certain assumptions and its quasi-optimality in comparison with the former algorithm. Furthermore, the computational efficiency and the convergence reliability of the latter algorithm are demonstrated through numerical examples.

The maximum flow problem is one of the classic combinatorial optimization problems with many applications, such as electrical powers, traffics, communications, computer networks and logistics. The problem is to find a flow of maximum value on a network from a source to a sink. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching the nodes and edges implicitly. For the maximum flow problem in networks, we present the symbolic algebraic decision diagram (ADD) formulation and symbolic algorithms. The augmenting-path-based symbolic algorithm is the combination of the Gabow's scaling algorithm with the binary decision diagram (BDD)-based symbolic algorithm for the maximum flow in 0–1 networks. The Karzanov's algorithm is implemented implicitly, resulting in the preflow-based symbolic algorithm (PSA), in which the vertices on each layer of the layered networks are partitioned by the vertex's input preflow and the total capacity of its outgoing edges, and the edges to push or pull the preflow are selected in terms of the priority function. The improved PSA is developed by integrating the heuristic of Traff's algorithm and the assistant layered networks into the PSA. The symbolic algorithms do not require explicit enumeration of the nodes and edges, and therefore can handle large-scale networks.

This paper presents an algorithm for finding a global minimum of a multimodal, multivariate and nondifferentiable function. The algorithm is a modification to the new version of the Price's algorithm given in Brachetti et al. [J. Global Optim. 10, 165---184 (1997)]. Its distinguishing features include: (1) The number-theoretic method is applied to generate the initial population so that the points in the initial population are uniformly scattered, and therefore the algorithm could explore uniformly the region of interest at the initial iteration; (2) The simplified quadratic approximation with the three best points is employed to improve the local search ability and the accuracy of the minimum function value, and to reduce greatly the computational overhead of the algorithm. Two sets of experiments are carried out to illustrate the efficiency of the number-theoretic method and the simplified quadratic model separately. The proposed algorithm has also been compared with the original one by solving a wide set of benchmark problems. Numerical results show that the proposed algorithm requires a smaller number of function evaluations and, in many cases, yields a smaller or more accurate minimum function value. The algorithm can also be used to deal with the medium size optimization problems.

This paper presents a continuation and optimization based algorithm to detect power flow unsolvability. In addition, the algorithm obtains the power flow solution, if it exists, no matter how ill-conditioned the power system is. The proposed algorithm is based on the parameterization of the distance from the starting point to the real power flow to be solved, using a convergence margin. The performance of the algorithm is illustrated considering an highly loaded scenario of the operation of the Spanish power system.

An algorithm for the longest-path subproblem in the network algorithm of Mehta and Patel for Fisher’s exact test in two-way contingency tables is proposed. Use of the primaldual technique for convex-cost flows and the maximum likelihood estimate makes the algorithm more efficient in time and space than existing algorithms.

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 min-cost 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 ${\mathrm O}(n^3)$ cuts per scaling phase, where $n$ is the number of nodes. Furthermore, we 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.

First, we present two classes of sequential algorithms for minimum flow problem: decreasing path algorithms and preflow algorithms. Then we describe another approach of the minimum flow problem, that consists of applying any maximum flow algorithm in a modified network. In section 5 we present several parallel preflow algorithms that solve the minimum flow problem. Finally, we present an application of the minimum flow problem.

Quite recently, Fujishige (Oper. Res. Lett. 31 (2003) 176-178) developed a weakly polynomial-time algorithm for the maximum flow problem by applying the maximum adjacency (MA) ordering technique to directed networks. In this note, we show that the algorithm is not strongly polynomial by giving a real-valued instance for which the algorithm does not terminate.

An algorithm called genetic shortest-path algorithm is presented to solve capacitated minimal Steiner tree problems in graphs with complex flows and arbitrary arc cost functions, but without negative cycles. Voltage constraint can also been taken into consideration by the algorithm. Hence, it can solve various power distribution system optimization problems with detailed mathematical models. In the proposed algorithm, a local optimization method based on shortest-path algorithm and heuristics is used to find the local optimums, in which the minimum cost objective and all constraints are considered and the specialties of the problems are made good use of. Genetic operations are only used to search the global optimum from the local optimums. Therefore, this algorithm overcomes the disadvantage of general genetic algorithm in local searching. An example for distribution system planning problem with large scale is given to demonstrate the power of the algorithm.

We propose two flow control algorithms for networks with multiple paths between each source-destination pair. Both are distributed algorithms over the network to maximize aggregate source utility. Algorithm 1 is a first order Lagrangian method applied to a modified objective function that has the same optimal solution as the original objective function but has a better convergence property. Algorithm 2 is based on the idea that, at optimality, only paths with the minimum price carry positive flows, and naturally decomposes the overall decision into flow control (determines total transmission rate based on minimum path price) and routing (determines how to split the flow among available paths). Both algorithms can be implemented as simply a source-based mechanism in which no link algorithm nor feedback is needed. We present numerical examples to illustrate their behavior.

An incremental algorithm may yield an enormous computational time saving to solve a network flow problem. It updates the solution to an instance of a problem for a unit change in the input. In this paper we have proposed an efficient incremental implementation of maximum flow problem after inserting an edge in the network G. The algorithm has the time complexity of O((Δn)2m), where Δn is the number of affected vertices and m is the number of edges in the network. We have also discussed the incremental algorithm for deletion of an edge in the network G.

The network problems multi-terminal flow, minimum cost flow, and multicommodity flow are fundamental to network design and performance evaluation. The first two can be solved exactly by the Gomory-Hu and the out-of-kilter algorithms, respectively. The former is a combinatorial algorithm on graph cuts, and the latter a primal-dual method. Multi-commodity flow is NP-complete for integer flows, and even for real-valued flows it is rather complex. Even if it in principle can be solved by linear programming, the number of constraints grow very fast with the problem size. An approximate method is therefore presented, which is suitable also for large problem instances.

This paper is concerned with an efficient global optimization algorithm for solving a kind of fractional program problem(P), whose objective and constraints functions are all defined as the sum of ratios generalized polynomial functions. The proposed algorithm is a combination of the branch-and-bound search and two reduction operations, based on an equivalent monotonic optimization problem of(P). The proposed reduction operations specially offer a possibility to cut away a large part of the currently investigated region in which the global optimal solution of(P)does not exist, which can be seen as an accelerating device for the solution algorithm of(P). Furthermore, numerical results show that the computational efficiency is improved by using these operations in the number of iterations and the overall execution time of the algorithm, compared with other methods. Additionally, the convergence of the algorithm is presented, and the computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem(P)provided that the number of variables is not too large.

Abstract Some special cases of the Maximum Flow Problem under nonnegative lower bounds on arc flows are investigated. Assuming that at least one of these bounds is strictly positive, a class of problems has been identified for which a single phase algorithm suffices to prove optimality. This situation contrasts with the two phases algorithm which is commonly used in the literature to solve these problems. In spite of having the same overall complexity of the two phases algorithm, the single phase one appears to be, computationally speaking, more attractive.

We present an algorithm for solving the minimum-cut problem on closure graphs without maintaining flow values. The algorithm is based on an optimization algorithm for the open-pit mining problem that was presented in 1964 (and published in 1965) by Lerchs and Grossmann. The Lerchs—Grossmann algorithm (LG algorithm) solves the maximum closure which is equivalent to the minimum-cut problem. Yet, it appears substantially different from other algorithms known for solving the minimum-cut problem and does not employ any concept of flow. Instead, it works with sets of nodes that have a natural interpretation in the context of maximum closure in that they have positive total weight and are closed with respect to some subgraph. We describe the LG algorithm and study its features and the new insights it reveals for the maximum-closure problem and the maximum- flow problem. Specifically, we devise a linear time procedure that evaluates a feasible flow corresponding to any iteration of the algorithm. We show that while the LG algorithm is pseudopolynomial, our variant algorithms have complexity of O(mn log n), where n is the number of nodes and m is the number of arcs in the graph. Modifications of the algorithm allow for efficient sensitivity and parametric analysis also running in time O(mn log n). © 2001 John Wiley & Sons, Inc.

Goldberg and Rao recently proposed the blocking flow method based on a binary length function to obtain a better algorithm for the maximum flow problem. The previous algorithms based on the blocking flow method proposed by Dinic use the unit length function: every residual edge is of length 1. In this paper, we survey properties of the distance function defined by a length function and give an overview on the representative maximum flow algorithms proposed so far in a systematic way by utilizing these properties. Among them are included two new algorithms: the Goldberg-Rao algorithm which finds a maximum flow on an integral capacity network N of n vertices and m edges in ~(min{ml/~, n2/3}7n log(n2/m) log U) time, where U is the maximum edge capacity of N, and the Karger-Levine algorithm which finds a maximum flow on an undirected network N with unit capacity and no parallel edges in 0(m + nv3/2) time, where v is the value of a maximum flow of N.

In this paper, we present a new polynomial time algorithm for solving the minimum cost network flow problem. This algorithm is based on Edmonds-Karp's capacity scaling and Orlin's excess scaling algorithms. Unlike these algorithms, our algorithm works directly with the given data and original network, and dynamically adjusts the scaling factor between scaling phases, so that it performs at least one flow augmentation in each phase. Our algorithm has a complexity of O(m(m+nlogn)log(B/(m+n))), where n is the number of nodes, m is the number of arcs, and B is the sum of the finite arc capacities and supplies in the network.

The minimum cost flow problem is to determine a least cost shipment of a commodity through a network G = (N, A) in order to satisfy demands at certain nodes from available supplies at other nodes. In this paper, we study a variant of the minimum cost flow problem where we are given a set R ⊆ A of arcs and require that each arc in R must carry the same amount of flow. This problem, which we call the simple equal flow problem, arose while modeling a water resource system management in Sardinia, Italy. We consider the simple equal flow problem in a directed network with n nodes, m arcs, and where all arc capacities and node supplies are integer and bounded by U. We develop several algorithms for the simple equal flow problem—the network simplex algorithm, the parametric simplex algorithm, the combinatorial parametric algorithm, the binary search algorithm, and the capacity scaling algorithm. The binary search algorithm solves the simple equal flow problem in O(log(nU)) applications of any minimum cost flow algorithm. The capacity scaling algorithm solves it in O(m(m + n logn) log (nU)) time, which is almost the same time needed to solve the minimum cost flow problem by the capacity scaling algorithm. These algorithms can be easily modified to obtain an integer solution of the simple equal flow problem.

The Stoer‐Wagner algorithm computes a minimum cut in a weighted undirected graph. The algorithm works inn 1 phases, wheren is the number of nodes ofG. Each phase takes time O.mCn logn/ ,w herem is the number of edges ofG, and computes a pair of vertices s and t and a minimum cut separatings and t . We show how to extend the algorithm such that each phase also computes a maximum flow from s tot . The flow is computed in O.m/ additional time and certifies the cut computed in the phase. © 1999 Elsevier Science B.V. All rights reserved. We give a correctness certificate for the min-cut algorithm of Stoer and Wagner [6]. This algorithm refines the algorithm of Nagamochi and Ibaraki [5,4]; it has the same time complexity, but is simpler. The algorithm is deterministic. There are faster randomized algorithms [2,3]. No certificates are known for the randomized algorithms. Let G be an undirected graph with n vertices and m edges. The algorithm of Stoer and Wagner works inn 1 phases. In each phase it identifies two vertices and a minimum cut separating them and then collapses the two vertices into one. The global min-cut is the minimum cut found in any phase. Each phase runs in time O.mCn logn/. We extend the algorithm such that it also finds a flow equal to the min-cut in each phase. The additional time required to find the flow is O.m/ in each phase.