Lexicographically Maximum Flows under an Arc Interdiction

Network interdiction problem arises when an unwanted agent attacks the network system to deteriorate its transshipment efficiency. Literature is flourished with models and solution approaches for the problem. This paper considers a single commodity lexicographic maximum flow problem on a directed network with capacitated vertices to study two network flow problems under an arc interdiction. In the first, the objective is to find an arc on input network to be destroyed so that the residual lexicographically maximum flow is lexicographically minimum. The second problem aims to find a flow pattern resulting lexicographically maximum flow on the input network so that the total residual flow, if an arc is destroyed, is maximum. The paper proposes strongly polynomial time solution procedures for these problems.

Maximum Network Flow Algorithms

The aim of the maximum network flow problem is to push as much flow as possible between two special vertices, the source and the sink satisfying the capacity constraints. For the solution of the maximum flow problem, there exists a number of algorithms. The existing algorithms can be divided into two families. First, augmenting path algorithms that satisfy the conservation constraints at intermediate vertices and the second preflow push relabel algorithms that violates the conservation constraints at the intermediate vertices resulting incoming flow more than outgoing flow.In this paper, we study different algorithms that determine the maximum flow in the static and dynamic networks.

Parallel Whale Optimization Algorithm for Maximum Flow Problem

Maximum Flow Problem (MFP) is considered as one of several famous problems in directed graphs. Many researchers studied MFP and its applications to solve problems using different techniques. One of the most popular algorithms that are employed to solve MFP is Ford-Fulkerson algorithm. However, this algorithm has long run time when it comes to application with large data size. For this reason, this study presents a parallel whale optimization (PWO) algorithm to get maximum flow in a weighted directed graph. The PWO algorithm is implemented and tested on datasets with different sizes. The PWO algorithm achieved up to 3.79 speedup on a machine with 4 processors.