Parametric Multiroute Flow and Its Application to Robust Network with $k$ Edge Failures

Abstract

In this work, we investigate properties of the function taking the real value $h$ to the max $h$-route flow value, and apply the result to solve robust network flow problems. We show that the function is piecewise hyperbolic, and modify a parametric optimization technique, the ES algorithm, to find this function. The running time of the algorithm is $O(\lambda mn)$, when $\lambda$ is a source-sink edge connectivity of our network, $m$ is the number of links, and $n$ is the number of nodes. We can use the result from that algorithm to solve two max-flow problems against $k$ edge failures, referred to as max-MLA-robust flow and max-MLA-reliable flow. When $h$ is optimally chosen from the function, we show that the max-$h$-route flow is an exact solution of both problems for graphs in a specific class. Our numerical experiments show that $98\,\%$ of random graphs generated in the experiment are in that specific class. Given a parametric edge $e$, we also show that the function taking the capacity of $e$ to the max-$h$-route flow value is linear piecewise. Hence we can apply our modified ES algorithm to find that function in $O(h^2mn)$.