Probabilistic tree-based representation for solving minimum cost integer flow problems with nonlinear non-convex cost functions

The minimum cost flow problem (MCFP) is the most generic variation of the network flow problem which aims to transfer a commodity throughout the network to satisfy demands. The problem size (in terms of the number of nodes and arcs) and the shape of the cost function are the most critical factors when considering MCFPs. Existing mathematical programming techniques often assume the cost functions to be linear or convex. Unfortunately, the linearity and convexity assumptions are too restrictive for modelling many real-world scenarios. In addition, many real-world MCFPs are large-scale, with networks having a large number of nodes and arcs. In this paper, we propose a probabilistic tree-based genetic algorithm (PTbGA) for solving large-scale minimum cost integer flow problems with nonlinear non-convex cost functions. We first compare this probabilistic tree-based representation scheme with the priority-based representation scheme, which is the most commonly-used representation for solving MCFPs. We then compare the performance of PTbGA with that of the priority-based genetic algorithm (PrGA), and two state-of-the-art mathematical solvers on a set of MCFP instances. Our experimental results demonstrate the superiority and efficiency of PTbGA in dealing with large-sized MCFPs, as compared to the PrGA method and the mathematical solvers.     

Joint gateway selection,transmission slot assignment,routing and power control for wireless mesh networks

Wireless mesh networks (WMNs) provide cost effective solutions for setting up a communications network over a certain geographic area. In this paper, we study strategic problems of WMNs such as selecting the gateway nodes along with several operational problems such as routing, power control, and transmission slot assignment. Under the assumptions of the physical interference model and the tree-based routing restriction for traffic flow, a mixed integer linear programming (MILP) formulation is presented, in which the objective is to maximize the minimum service level provided at the nodes. A set of valid inequalities is derived and added to the model in an attempt to improve the solution quality. Since the MILP formulation becomes computationally infeasible for larger instances, we propose a heuristic method that is aimed at solving the problem in two stages. In the first stage, we devise a simple MILP problem that is concerned only with the selection of gateway nodes. In the second stage, the MILP problem in the original formulation is solved by fixing the gateway nodes from the first stage. Computational experiments are provided to evaluate the proposed models and the heuristic method.     

An efficient network flow code for finding all minimum cost s–t cutsets

Cutset algorithms have been well documented in the operations research literature. A directed graph is used to model the network, where each node and arc has an associated cost to cut or remove it from the graph. The problem considered in this paper is to determine all minimum cost sets of nodes and/or arcs to cut so that no directed paths exist from a specified source node s to a specified sink node t. By solving the dual maximum flow problem, it is possible to construct a binary relation associated with an optimal maximum flow such that all minimum cost s–t cutsets are identified through the set of closures for this relation. The key to our implementation is the use of graph theoretic techniques to rapidly enumerate this set of closures. Computational results are presented to suggest the efficiency of our approach.Scope and purposeThis paper describes the technical details of a network flow algorithm used to find all minimum cost s–t cutsets in any network topology. The motivation for this work was to provide additional automated analysis capability to a military network targeting system. Specifically, the problem is to identify a minimum cost set of nodes and/or arcs that when removed from the network, will disconnect a selected pair of origin–destination nodes. Algorithms for solving this problem are well understood, with an active research thrust in both the operations research and computer science academic communities in developing more efficient algorithms for larger networks. The main contribution of this paper is in extending these algorithms to quickly find all minimum cost cutset solutions. The implementation described in this paper outperformed conventional methods by several orders of magnitude on networks having thousands of nodes and arcs, with empirical solution times that grew linearly with the network size. The results translate to a real-time cutset analysis capability to support military targeting applications.     

Decomposition's Dantzig–Wolfe applied to fuzzy multicommodity flow problems

We present, in this paper, a method for solving linear programming problems with fuzzy costs based on the classical method of decomposition's Dantzig–Wolfe. Methods using decomposition techniques address problems that have a special structure in the set of constraints. An example of such a problem that has this structure is the fuzzy multicommodity flow problem. This problem can be modeled by a graph whose nodes represent points of supply, demand and passage of commodities, which travel on the arcs of the network. The objective is to determine the flow of each commodity on the arcs, in order to meet demand at minimal cost while respecting the capacity constraints of the arcs and the flow conservation constraints of the nodes. Using the theory of fuzzy sets, the proposed method aims to find the optimal solution, working with the problem in the fuzzy form during the resolution procedure.     

Multiple allocation hub-and-spoke network design under hub congestion

The multiple allocation hub-and-spoke network design under hub congestion problem is addressed in this paper. A non-linear mixed integer programming formulation is proposed, modeling the congestion as a convex cost function. A generalized Benders decomposition algorithm has been deployed and has successfully solved standard data set instances up to 81 nodes. The proposed algorithm has also outperformed a commercial leading edge non-linear integer programming package. The main contribution of this work is to establish a compromise between the transportation cost savings induced by the economies of scale exploitation and the costs associated with the congestion effects.     

Maximal matching polytope in trees

Given a weighted simple graph, the minimum weighted maximal matching (MWMM) problem is the problem of finding a maximal matching of minimum weight. The MWMM problem is NP-hard in general, but is polynomial-time solvable in some special classes of graphs. For instance, it has been shown that the MWMM problem can be solved in linear time in trees when all the edge weights are equal to one. In this paper, we show that the convex hull of the incidence vectors of maximal matchings (the maximal matching polytope) in trees is given by the polytope described by the linear programming relaxation of a recently proposed integer programming formulation. This establishes the polynomial-time solvability of the MWMM problem in weighted trees. The question of whether or not the MWMM problem can be solved in linear time in weighted trees is open.     

Cycle Flow Formulation of Optimal Network Flow Problems and Respective Distributed Solutions

In this paper, we use the cycle basis from graph theory to reduce the size of the decision variable space of optimal network flow problems by eliminating the aggregated flow conservation constraint. We use a minimum cost flow problem and an optimal power flow problem with generation and storage at the nodes to demonstrate our decision variable reduction method. The main advantage of the proposed technique is that it retains the natural sparse/decomposable structure of network flow problems. As such, the reformulated problems are still amenable to distributed solutions. We demonstrate this by proposing a distributed alternating direction method of multipliers (ADMM) solution for a minimum cost flow problem. We also show that the communication cost of the distributed ADMM algorithm for our proposed cycle-based formulation of the minimum cost flow problem is lower than that of a distributed ADMM algorithm for the original arc-based formulation.      

Rough‐fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi‐objective rough‐fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough‐fuzzy variables. The proposed model is formulated by using rough‐fuzzy chance‐constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi‐objective solution technique, the epsilon‐constraint method and two multi‐objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA‐II) and (b) multi‐objective cross‐generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA‐II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.     

A virtual network mapping algorithm based on integer programming

The virtual network （VN） embedding/mapping problem is recognized as an essential question of network virtualization. The VN embedding problem is a major challenge in this field. Its target is to efficiently map the virtual nodes and virtual links onto the substrate network resources. Previous research focused on designing heuristic-based algorithms or attempting two-stage solutions by solving node mapping in the first stage and link mapping in the second stage. In this study, we propose a new VN embedding algorithm based on integer programming. We build a model of an augmented substrate graph, and formulate the VN embedding problem as an integer program with an objective function and some constraints. A factor of topology-awareness is added to the objective function. The VN embedding problem is solved in one stage. Simulation results show that our algorithm greatly enhances the acceptance ratio, and increases the revenue/cost （R/C） ratio and the revenue while decreasing the cost of the VN embedding problem.     

Dinkelbach NCUT: An Efficient Framework for Solving Normalized Cuts Problems with Priors and Convex Constraints

In this paper, we propose a novel framework, called Dinkelbach NCUT (DNCUT), which efficiently solves the normalized graph cut (NCUT) problem under general, convex constraints, as well as, under given priors on the nodes of the graph. Current NCUT methods use generalized eigen-decomposition, which poses computational issues especially for large graphs, and can only handle linear equality constraints. By using an augmented graph and the iterative Dinkelbach method for fractional programming (FP), we formulate the DNCUT framework to efficiently solve the NCUT problem under general convex constraints and given data priors. In this framework, the initial problem is converted into a sequence of simpler sub-problems (i.e. convex, quadratic programs (QP’s) subject to convex constraints). The complexity of finding a global solution for each sub-problem depends on the complexity of the constraints, the convexity of the cost function, and the chosen initialization. However, we derive an initialization, which guarantees that each sub-problem is a convex QP that can be solved by available convex programming techniques. We apply this framework to the special case of linear constraints, where the solution is obtained by solving a sequence of sparse linear systems using the conjugate gradient method. We validate DNCUT by performing binary segmentation on real images both with and without linear/nonlinear constraints, as well as, multi-class segmentation. When possible, we compare DNCUT to other NCUT methods, in terms of segmentation performance and computational efficiency. Even though the new formulation is applied to the problem of spectral graph-based, low-level image segmentation, it can be directly applied to other applications (e.g. clustering).     

A biased random-key genetic algorithm for wireless backhaul network design

This paper describes a biased random-key genetic algorithm for a real-world wireless backhaul network design problem. This is a novel problem, closely related to variants of the Steiner tree problem and the facility location problem. Given a parameter h, we want to build a forest where each tree has at most h hops from the demand nodes, where traffic originates, to the root nodes where each tree is rooted. Candidate Steiner nodes do not have any demand but represent locations where we can install cellsites to cover the traffic and equipment to backhaul the traffic to the cellular core network. Each Steiner node can cover demand nodes within a given distance, subject to a capacity constraint. The aggregate set of constraints may make it impossible to cover or backhaul all demands. A revenue function computes the revenue associated with the total amount of traffic covered and backhauled to the root nodes. The objective of the problem is to build a forest that maximizes the difference between the total revenue and the cost associated with the installed equipment. Although we will have a forest when we consider only the backhaul links and root nodes, the addition of demand vertices can induce undirected cycles, resulting in a directed acyclic graph. We consider instances of this problem with several additional constraints that are motivated by the requirements of real-world telecommunication networks.     

Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.     

一类实际网络中的最小截算法

讨论了节点和边都有容量限制的无向平面网络中的两点间的最小截问题.传统方法是把节点和边都有容量的网络中的最小截问题转化为只有边有容量的问题,但该方法用在平面网络时不能保持网络的平面性,因此网络的平面性不能得到利用.使用传统方法的计算时间为O(n²logn)(其中n为网络的节点数).给出了可以充分利用网络平面性的方法.对源和汇共面的s-t平面网络,把最小截问题转化为平面图上两点间的最短路径问题,从而可以得到O(n)时间的算法;对一般的平面网络,给出了新的将节点和边都有容量的问题转化为仅边有容量问题的方法,这种转化方法不破坏网络的平面性,从而可以利用平面网络中仅边有容量问题的计算方法,使原问题在O(nlogn)时间内获得解决.     

求解含非线性参数的QoS路由的启发式算法

具有非线性参数的QoS路由分为含有非线性约束条件的QoS路由和含有非线性优化目标的QoS路由两类，它们都是NP问题．提出了两种启发式算法求解这两类QOS路由优化问题问题．对第一类问题，求解去掉非线性约束条件后的优化问题．如果找到的解满足非线性约束条件，则该解是最优解；否则在优化问题中添加一个新的线性约束，将已得到的解去掉，反复下去就可得到最终解．对第二类问题，将非线性优化目标换为约束条件中的线性参数，求解此优化模型，如果有解，则记录此时对应的非线性目标值．而后增加一个新的线性约束，去掉刚才得到的解，比较两次得到的非线性目标值，保留最小值．如果得到的解不满足该线性参数的约束条件，则算法结束；否则继续迭代．证明了两种算法的收敛性，并且时间复杂性为近似多项式时间．计算实例表明了算法的有效性．     

Minimal cost replication of dynamic Web contents under flat update delivery

Dynamic Web contents are generated by running application programs on base data which often change frequently. Geographically replicating the applications that construct these contents (including the programs and the related data they access) is an effective approach to improve their access latency. To maintain the freshness of an object replica, the new version of the object either has to be fetched from remote servers or be reconstructed locally when the origin copy is updated. We present a theoretical study on geographical replication of dynamic Web contents with the objective of minimizing the consistency management costs in terms of update transfers and object reconstruction. The dependencies among dynamic objects and base data are modeled as a directed acyclic graph. We formulate the minimum cost replication problem under a flat framework of update delivery. The problem is solved by first transforming it into a minimum cut problem in a flow network. A polynomial-time algorithm is then proposed to compute the optimal replication strategy which designates where each object should be replicated and how to keep the replicas up-to-date.     

纹理合成中基于非标量距离度量的Graph Cut方法

在逐块纹理合成中,Graph Cut方法被广泛用于优化块间重叠区域的像素取值。传统Graph Cut方法采用的累积距离度量,使得切割路径趋于走捷径而穿过高误差区域。针对此问题,提出了一种基于非标量距离度量的GraphCut方法。提出了一种基于该度量的高效最小割算法,并证明了其最优性;讨论了改进Graph Cut方法的规则性问题,并给出了解决方法。实验结果表明,由改进的Graph Cut方法所得到的切割路径更加曲折和平滑,块边界隐蔽性更强。     

Solving the Freight Car Flow Problem to Optimality

A pervasive problem in freight railroad operations is to determine a feasible flow of cars to meet the required demands within a certain period of time. In this work we present a method to determine an optimal flow of loaded and empty cars in order to maximize profits, revenue or tonnage transported, given the schedule of the trains, together with their traction capacities. We propose an integer multicommodity flow model for the problem whose linear relaxation leads to very good upper bounds — at the cost of using a very large number of variables and constraints. In order to turn this model into a practical tool, we apply a preprocessing phase that may reduce its size by two or three orders of magnitude. The reduced model can then be solved by standard integer program packages with little, if any, branching effort. Computational results on real instances of the largest Latin American railroad freight company are reported. The product that resulted from this research is already in use at that company.     

Minimal communication cost software construction in the Internet environment

This paper investigates the issue of building software in the Internet environment, where local area network (LAN) based systems are interconnected by links with different bandwidth and do not share file systems. The software is modeled as a directed acyclic graph. Each node in the graph represents a logical step in processing the software while the edges describe the order of execution. The problem is to construct the software at a particular LAN with minimum Internet communication cost. An optimal polynomial algorithm, SOFTCON, with time complexity is presented, where and are the number of nodes and edges in the graph describing the software respectively, is the number of LANs in the Internet environment, and is the time complexity of the network flow algorithm on the flow network with nodes and edges transformed from the directed acyclic graph of the software. Received: 6 December 1995 / 1 May 1996     

求解最小费用最大流问题的信念传播算法

最小费用最大流问题是一种组合优化问题,在经济、工业等领域具有重要研究意义和应用价值.针对部分最小费用最大流问题求解算法效率较低的情况,依据最小费用最大流问题的线性规划方程,将问题模型映射为对应因子图模型,改进描述函数,给出迭代方程,设计了求解最小费用最大流问题的信念传播算法.利用迭代方程优先对最大可行流特征值进行收敛计算,得到最大流,设置最大流阈值,在此基础上进行最小费用计算,从而求得问题最优解.最后选取若干带权有向图模型进行数值实验,验证了算法的可行性及有效性,且算法在求解效率上优于部分算法.     

Designing cost-effective content distribution networks

In this paper, we present a novel technique for the problem of designing a Content Distribution Network (CDN), which is a technology used to efficiently distribute electronic content throughout an existing IP network. Our design proposal consists of jointly deciding on (i) the number and placement of proxy servers on a given set of potential nodes, (ii) replicating content on the proxy servers, and (iii) routing the requests for the content to a suitable proxy server such that the total cost of distribution is minimized. We model the problem using a nonlinear integer programming formulation. The novelty of the proposed formulation lies in simultaneously addressing three interdependent problems in this context as well as explicitly representing the distribution structure of a CDN through the objective function. We offer a linearization for the model, develop an exact solution procedure based on Benders’ decomposition and also utilize a variant of this procedure to accelerate the algorithm. In addition, we provide a fast and efficient heuristic that can be used to obtain near-optimal solutions to the problem. Finally, the paper concludes with computational results showing the performance of the decomposition procedure and the heuristic algorithm on randomly generated Internet topologies.