Parametric multiroute flow and its application to multilink-attack network

Abstract

We investigate variants of the max-flow problem in a network under k attacks. The network interdiction problem is to find the minimum max-flow value among (mk) networks that can be obtained by deleting each set of k links. The adaptive network flow problem is to find a flow of the network such that the flow value is maximum against any set of k links attack, when deleting the corresponding flow to those k links in the original flow. First, we prove that max-(k+1)-route flow is a (k+1)-approximation for both problems. Also, we develop a polynomial-time heuristic algorithm for both cases, called the iterative multiroute flow. Then in a second phase, we investigate properties of the function taking the real value h to the max h-route flow value, and apply the result to solve both of the problems. We show that the function is piecewise hyperbolic, and modify a standard parametric optimization technique to find this function. The running time of the algorithm is O(λT), when λ is a source–sink edge connectivity of our network and T the computation time of a max-flow algorithm. We show that for some instances, when h is optimally chosen, the max- h-route flow is an exact solution for both problems.