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

Abstract

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.

Network Analysis via Pseudo-Boolean Functions and Boole-Shannon Expansion

“…A Boolean function, also known as a Switching function, is a type of mapping that takes as input a combination of 𝑛 binary digits (either 0 or 1) and produces a single output digit that is also binary (either 0 or 1), {0, 1} 𝑛 → {0, 1}. In other words, 𝑆(𝑿) represents a unique combination of 0's and 1's for every possible combination of 𝑛 binary digit [3,7,[9][10][11]13]. However, a pseudo-switching (pseudo-Boolean) function 𝐶(𝑿) is a mapping {0, 1} 𝑛 → 𝑅 where 𝑅 is the field of real numbers, i.e., 𝐶(𝑿) assigns a real number to each of the 2 𝑛 possible 𝑿 values.…”

“…However, a pseudo-switching (pseudo-Boolean) function 𝐶(𝑿) is a mapping {0, 1} 𝑛 → 𝑅 where 𝑅 is the field of real numbers, i.e., 𝐶(𝑿) assigns a real number to each of the 2 𝑛 possible 𝑿 values. For binary capacitated networks and other applications, pseudo-Boolean functions play an essential role [2,3,7,[9][10][11]13]. This section discusses pseudo-switching functions briefly and provides some ideas on their utility in the analysis of binary capacitated networks [13].…”

“…and may be obtained directly (on a one-to-one basis) from 𝐶 𝑖𝑗 (𝑿) (s-o-p) by substituting the component reliabilities 𝑝 𝑙 = E{𝑋 𝑙 } and 𝑞 𝑙 = 𝐸{𝑋 𝑙 } , for the corresponding Boolean inputs 𝑋 𝑙 , and 𝑋 𝑙 . The resultant polynomial representation has traditionally been utilized to handle pseudo-Boolean functions [3,7,[9][10][11]13].…”

“…Many novel approaches for studying a capacitated network have been developed over the past few decades [1][2][3][4][5][6][7][8][9][10][11][12][13]. A broad class of these approaches is based on the repeated application of Bayesian decomposition to the network graph [14][15][16], or equivalently, of the Boole-Shannon expansion [17] to the switching (Boolean) function of the network success or network failure [18][19][20][21].…”

“…The Total Probability Theorem [30][31][32] in the probability domain and the Factoring Theorem in the "Graph Domain" [15] are both equivalent to the Boole-Shannon Expansion in the Boolean domain [21]. The Variable-Entered Karnaugh Map (VEKM) serves as a powerful visual tool for implementing this expansion [33,[7][8][9][10][11].…”