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Abstract:
Flows over time (also called dynamic flows)
generalize standard network flows by introducing
an element of time. They naturally model problems where
travel and transmission are not instantaneous. Solving these problems
raises issues that do not arise in standard network flows.
One issue is the question of storage of flow at intermediate nodes.
In most applications
(such as, e.g., traffic routing, evacuation planning,
telecommunications etc.), intermediate storage is limited, undesired,
or prohibited.
The minimum cost flow over time problem is NP-hard. In this paper we
1)~prove that the minimum cost flow over time never requires storage;
2)~provide the first approximation scheme for minimum cost flows over
time that does not require storage; 3)~provide the first approximation
scheme for minimum cost flows over time that meets hard cost
constraints, while approximating only makespan.
Our approach is based on a condensed variant of time-expanded
networks. It also yields fast approximation schemes with simple
solutions for the quickest multicommodity flow problem.
Finally, using completely different techniques, we describe a very
simple capacity scaling FPAS for the minimum cost flow over time
problem when costs are proportional to transit times.
The algorithm builds upon our observation about the
structure of optimal solutions to this problem:
they are universally quickest flows. Again, the FPAS
does not use intermediate node storage.
In contrast to the preceding algorithms that use a
time-expanded network, this FPAS runs directly on the original
network.
