A Review of Flow-Capacitated Networks: Algorithms, Techniques and Applications

This paper presents a review of flow network concepts, including definition of some graph-theoretic basics and a discussion of network flow properties. It also provides an overview of some crucial algorithms used to solve the maximum-flow problem such as the Ford and Fulkerson algorithm (FFA), supplemented with alternative solutions, together with the essential terminology for this algorithm. Moreover, this paper explains the max-flow min-cut theorem in detail, analyzes the concepts behind it, and provides some examples and their solutions to demonstrate this theorem. As a bonus, it expounds the reduction and transformation techniques used in a capacitated network. In addition, this paper reviews one of the popular techniques for analyzing capacitated networks, which is the “decomposition technique”. This technique is centered on conditioning a complicated network on the possible states of a keystone element or on the possible combinations of states of many keystone elements. Some applications of capacitated network problems are addressed based on each type of problem being discussed.

Checking Correctness of a Symbolic Reliability Expression for a Capacitated Network

Checking a symbolic reliability expression for a flow network is useful for detecting faults in hand derivations and for debugging computer programs. This checking can be achieved in a systematic way, though it may be a formidable task. Three exhaustive tests are given when a reliability system or network has a flow constraint. These tests apply to unreliability and reliability expressions for non-coherent as well as coherent systems, and to cases when both nodes and branches are unreliable. Further properties of reliability expressions derived through various methods are discussed. All the tests and other pertinent results are proved and illustrated by examples.

A Tutorial Exposition of Various Methods for Analyzing Capacitated Networks

In order to assess the performance indexes of some practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems, we propose various types of techniques for analyzing a capacitated network. These include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method.

Reliability Evaluation of Multi-State Flow Networks Via Map Methods

This paper examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks. These two methods involve two Karnaugh map versions, namely the Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM). These two versions are crucial in providing not only the visual insight necessary to write better future software but also adequate means of verifying such software. We assess these two versions of map methods versus the exhaustive search method, which guarantees conceptual clarity at the expense of lack of computational efficiency. Our target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which we consider flow from a source node to a sink node in a capacitated network with a multistate capacity model for the links. Each network link has a varying capacity, which is assumed to exist in a mutually exclusive sense. The reliability of the system is wholly dependent on its ability to successfully transmit at least a certain required system flow from the source (transmitter) to the sink (receiver) station. The max-flow min-cut theorem is critical in obtaining all successful states. To demonstrate the proposed methods applicability, two demonstrative examples are given with ample details.

Maximum Capacity Path Interdiction Problem with Fixed Costs

This paper addresses a network optimization interdiction problem, called the maximum capacity path interdiction problem. The problem is a hierarchical game containing two players: one evader and one interdictor. In a capacitated network, the evader wants to find a simple path from his current position to a target point with maximum capacity to send his forces along it while the interdictor decreases arc capacities under a budget constraint to interdict the advance of the evader’s forces as much as possible. This paper studies the case that each arc has a fixed cost for decreasing its capacity. An algorithm is proposed to solve the problem in strongly polynomial time. Computational experiments on two real-world datasets guarantee the efficiency and accuracy of the algorithm.