Faster feasibility for dynamic flows and transshipments on temporal networks

Abstract

In this paper we study flow problems on temporal networks, where edge capacities and travel times change over time. We consider a network with $n$ nodes and $m$ edges where the capacity and length of each edge is a piecewise constant function, and use $\mu=\Omega(m)$ to denote the total number of pieces in all of the $2m$ functions. Our goal is to design exact algorithms for various flow problems that run in time polynomial in the parameter $\mu$. Importantly, the algorithms we design are strongly polynomial, i.e. have no dependence on the capacities, flow value, or the time horizon of the flow process, all of which can be exponentially large relative to the other parameters; and return an integral flow when all input parameters are integral. Our main result is an algorithm for checking feasibility of a dynamic transshipment problem on temporal networks -- given multiple sources and sinks with supply and demand values, is it possible to satisfy the desired supplies and demands within a given time horizon? We develop a fast ($O(\mu^3)$ time) algorithm for this feasibility problem when the input network has a certain canonical form, by exploiting the cut structure of the associated time expanded network. We then adapt an approach of \cite{hoppe2000} to show how other flow problems on temporal networks can be reduced to the canonical format. For computing dynamic transshipments on temporal networks, this results in a $O(\mu^7)$ time algorithm, whereas the previous best integral exact algorithm runs in time $\tilde O(\mu^{19})$. We achieve similar improvements for other flow problems on temporal networks.