Interdiction in network maximum flow and related problems: A survey

The present chapter deals with the analogy that exists between the problem of maximum flow in networks [15] and equilibrium problems for physical systems described in Secs.7.2, 7.4 and 7.5 of the present chapter. We shall discuss models which differ among themselves only in the mechanical properties that cause constraints. In the first model these constraints are perfectly rigid and in others they are elastic. The problem under discussion constitutes a particular class of linear programming problems, and provides another example illustrating the significance and importance of analogies. The models are so simple that anybody with a moderate knowledge of mechanics can understand them. These models lead on the one hand to simple physical interpretations of the theoretical results, and, on the other hand, to the methods for obtaining numerical solutions that follow from the theory of equilibrium.KeywordsPotential EnergyEquilibrium ProblemIntermediate NodeMaximum FlowDuality TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

: This paper develops methods for the exact computation of the distribution of the maximum flow and related quantities in a planar network with independent and exponentially distributed arc capacities. A continuous time Markov chain (CTMC) with upper triangular rate matrix and single absorbing state is equal to the value of maximum flow in the network. Recursive algorithms are developed for computing the distribution and moments of the maximum flow. Algorithms are also presented to compute the probability that a given cut is the minimum capacity cut in the network. The algorithms are efficient and computationally stable. Distribution of the maximum flow, given a minimum cut, is studied. Keywords include: Maximum flow; Stochastic networks; Multi-state reliability modeling; Markov chains.

In the article the problem of finding the maximum multiple flow in the network of any natural multiplicity k is studied. There are arcs of three types: ordinary arcs, multiple arcs and multi-arcs. Each multiple and multi-arc is a union of k linked arcs, which are adjusted with each other. The network constructing rules are described. The definitions of a divisible network and some associated subjects are stated. The important property of the divisible network is that every divisible network can be partitioned into k parts, which are adjusted on the linked arcs of each multiple and multi-arc. Each part is the ordinary transportation network. The main results of the article are the following subclasses of the problem of finding the maximum multiple flow in the divisible network. 1. The divisible networks with the multi-arc constraints. Assume that only one vertex is the ending vertex for a multi-arc in s network parts. In this case the problem can be solved in a polynomial time. 2. The divisible networks with the weak multi-arc constraints. Assume that only one vertex is the ending vertex for a multi-arc in k-1 network parts (1 ≤ s < k − 1) and other parts have at least two such vertices. In that case the multiplicity of the maximum multiple flow problem can be decreased to k - s. 3. The divisible network of the parallel structure. Assume that the divisible network component, which consists of all multiple arcs, can be partitioned into subcomponents, each of them containing exactly one vertex-beginning of a multi-arc. Suppose that intersection of each pair of subcomponents is the only vertex-network source x0. If k=2, the maximum flow problem can be solved in a polynomial time. If k ≥ 3, the problem is NP-complete. The algorithms for each polynomial subclass are suggested. Also, the multiplicity decreasing algorithm for the divisible network with weak multi-arc constraints is formulated.

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

Vulnerability importance measures toward resilience-based network design

An O(V cubed) algorithm is given for finding maximum flow in capacitated networks. THe algorithm is already well refenreces in literature and is being posted here to make it accessible to all in electronic form.

Chapter 5 - Efficient CUDA Algorithms for the Maximum Network Flow Problem

Minoux considered the maximum balanced flow problem, i.e. the problem of finding a maximum flow in a two-terminal network N=(V, A) with source s and sink t satisfying the constraint that any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where V is vertex set and A is arc set. Several efficient algorithms, so far, have been proposed for this problem. As a natural generalization of this problem we focus on the problem of maximizing the total flow value of a generalized flow in a network N=(V, A) with gains γ(a)>0 (a∈A) satisfying any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where γ(a)f(a) units arrive at the vertex w for each arc-flow f(a) (a≡(v, w)∈A) entering vertex v in a generalized flow in AF. We call it the generalized maximum balanced flow problem and if γ(a)=1 for any a∈A then it is a maximum balanced flow problem. The authors believe that no algorithms have been shown for this generalized version. Our main results are to propose two polynomial algorithms for solving the generalized maximum balanced flow problem. The first algorithm runs in O(mM(n, m, B')log B) time, where B is the maximum absolute value among integral values used by an instance of the problem, and M(n, m, B') denotes the complexity of solving a generalized maximum flow problem in a network with n vertices, and m arcs, and a rational instance expressed with integers between 1 and B'. In the second algorithm we combine a parameterized technique of Megiddo with one of algorithms for the generalized maximum flow problem, and show that it runs in O({M(n, m, B')}^2)time.

Aerial base station (ABS) is a promising solution for public safety as it can be deployed in coexistence with cellular networks to form a temporary communication network. However, the interference from the primary cellular network may severely degrade the performance of an ABS network. With this consideration, an adaptive dynamic interference avoidance scheme is proposed in this work for ABSs coexisting with a primary network. In the proposed scheme, the mobile ABSs can reconfigure their locations to mitigate the interference from the primary network, so as to better relay the data from the designated source(s) to destination(s). To this end, the single/multi-commodity maximum flow problems are formulated and the weighted Cheeger constant is adopted as a criterion to improve the maximum flow of the ABS network. In addition, a distributed algorithm is proposed to compute the optimal ABS moving directions. Moreover, the trade-off between the maximum flow and the shortest path trajectories is investigated and an energy-efficient approach is developed as well. Simulation results show that the proposed approach is effective in improving the maximum network flow and the energy-efficient approach can save up to 39% of the energy for the ABSs with marginal degradation in the maximum network flow.

This paper investigates the application of a minimum loss path finding algorithm to determine the maximum flow in generalized networks that are characterized by arc losses or gains. In these generalized network flow problems, each arc has not only a defined capacity but also a loss or gain factor, which must be taken into consideration when calculating the maximum achievable flow. This extension of the traditional maximum flow problem requires a more comprehensive approach, where the maximum amount of flow is determined by accounting for additional factors such as costs, varying arc capacities, and the specific loss or gain associated with each arc. This paper extends the classic Ford–Fulkerson algorithm, adapting it to iteratively identify source-to-sink (s − t) residual directed paths with minimum cumulative loss and generalized augmenting paths (GAPs), thus enabling the efficient computation of maximum flow in such complex networks. Moreover, to enhance the computational performance of the proposed algorithm, we conducted extensive studies on parallelization techniques using graphics processing units (GPUs). Significant improvements in the algorithm’s efficiency and scalability were achieved. The results demonstrate the potential of GPU-accelerated computations in handling real-world applications where generalized network flows with arc losses and gains are prevalent, such as in telecommunications, transportation, or logistics networks.

Minimum Cost Maximum Flow Problem (MCMFP) is to find the maximum flow with minimal total cost, i.e., the minimum cost maximum flow, in transportation network. In this paper, MCMFP is formulated using a bi-level programming model. As a solution method of the model, an algorithm named MCMF-A is proposed. The basic idea of the MCMF-A algorithm can be described as follows: first find a maximum flow without opposite flow in transportation network, next construct its residual network with regard to the maximum flow, then find a negative-cost cycle in the residual network by calling well-known Floyd Algorithm; if Floyd Algorithm finds that there is no negative-cost cycle in the residual network, then the maximum flow is minimum cost maximum flow, else adjust the maximum flow via the negative-cost cycle to get a new maximum flow with no opposite flow and less total transportation cost. The theory, on which the MCMF-A algorithm is based, is presented. The MCMF-A algorithm can find the optimal solution to MCMFP, and has good performance in the sense of being implemented on computer, computational time and required memory for computation. Numerical experiments demonstrate that the MCMF-A algorithm is an efficient and robust method to solve MCMFP, which can serve as an effective tool to solve other related optimization problems.

A water distribution system which consists of distribution network and pumping station is considered. The global energy performance and hydraulic power capacity of the water distribution network is analysed. The energetically maximum flows in pipes and networks are determined. The coefficient of output power efficiency and the surplus power factor of hydraulic systems are defined. The assemblage of characteristic curves of water distribution network is analysed from the aspect of energetically maximum flows in network.

Discrete and continuous time dynamic flow problems have been studied for decades. The purpose of the network flow problem is to find the maximum flow that can be sent from the source node to the destination node. Our aim is to review the general class of continuous time dynamic flow problems. We discuss about static cut and generalized dynamic cut, the latter one used to prove the maximum flow minimum cut theorem in continuous case.

This paper presents and solves the maximum throughput dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which the flow on each arc not only has an associated upper and lower bound but also an associated transit time. Flow is to be sent through the network in each period so as to satisfy the upper and lower bounds and conservation of flow at each node from some fixed period on. The objective is to maximize the throughput, the net flow circulating in the network in a given period, and this throughput is shown to be the same in each period. We demonstrate that among those flows with maximum throughput there is a flow which repeats every period. Moreover, a duality result shows the maximum throughput equals the minimum capacity of an appropriately defined cut. A special case of the maximum dynamic network flow problem is the problem of minimizing the number of vehicles to meet a fixed periodic schedule. Moreover, the elegant solution derived by Ford and Fulkerson for the finite horizon maximum dynamic flow problem may be viewed as a special case of the infinite horizon maximum dynamic flow problem and the optimality of solutions which repeat every period.