Minimum‐Peak‐Cost Flows Over Time

ABSTRACT Peak cost is a novel objective for flows over time that describes the amount of workforce necessary to run a system. We focus on minimizing peak costs in the context of maximum temporally repeated flows and formulate the corresponding MPC‐MTRF problem. First, we discuss the limitations that emerge when restricting the solution space to integral temporally repeated flows, which is motivated by practical applications. We show that, in general, MPC‐MTRF has an integrality gap of and an arbitrarily bad approximation ratio compared to general flows over time. We proceed with a complexity analysis for MPC‐MTRF and show that both the decision version and the optimization version of integral MPC‐MTRF are strongly ‐hard, even under strong restrictions. On the positive side, we identify two special cases that are solvable in polynomial time: unit‐cost series‐parallel networks and networks with time horizon at least twice as long as the longest path in the network with respect to the transit time. Moreover, in both cases, we provide an explicit algorithm that constructs an integral optimal solution.