Abstract
We introduce temporal flows on temporal networks [17, 19], i.e., networks the links of which exist only at certain moments of time. Such networks are ephemeral in the sense that no link exists after some time. Our flow model is new and differs from the “flows over time” model, also called “dynamic flows” in the literature. We show that the problem of finding the maximum amount of flow that can pass from a source vertex \({s}\) to a sink vertex \({t}\) up to a given time is solvable in Polynomial time, even when node buffers are bounded. We then examine mainly the case of unbounded node buffers. We provide a simplified static Time-Extended network (\(\mathrm {STEG}\)), which is of polynomial size to the input and whose static flow rates are equivalent to the respective temporal flow of the temporal network; using \(\mathrm {STEG}\), we prove that the maximum temporal flow is equal to the minimum temporal \({s}\text {-}{t}\) cut. We further show that temporal flows can always be decomposed into flows, each of which moves only through a journey, i.e., a directed path whose successive edges have strictly increasing moments of existence. We partially characterise networks with random edge availabilities that tend to eliminate the \({s}\rightarrow {t}\) temporal flow. We then consider mixed temporal networks, which have some edges with specified availabilities and some edges with random availabilities; we show that it is #P-hard to compute the tails and expectations of the maximum temporal flow (which is now a random variable) in a mixed temporal network.
This work was partially supported by (i) the School of EEE and CS and the NeST initiative of the University of Liverpool, (ii) the NSERC Discovery grant, (iii) the Polish National Science Center grant DEC-2011/02/A/ST6/00201, and (iv) the FET EU IP Project MULTIPLEX under contract No. 317532.
Due to lack of space, an extended literature review and all missing proofs can be found in the full version of this paper at http://arxiv.org/abs/1606.01091 [2].
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Notes
- 1.
The first “dynamic” term refers to the dynamic nature of the underlying graph.
- 2.
We choose an even integer to simplify the calculations in the remainder of the paper. However, with careful adjustments, the results would still hold for an arbitrary integer.
- 3.
We choose an even integer to simplify the calculations. However, with careful adjustments in the calculations, the results would still hold for an arbitrary integer.
- 4.
\(\{0,1\}^* = \cup _{n \ge 0} \{0,1\}^n\), where \(\{0,1\}^n\) is the set of all strings (of bits 0, 1) of length n.
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Akrida, E.C., Czyzowicz, J., Gąsieniec, L., Kuszner, Ł., Spirakis, P.G. (2017). Temporal Flows in Temporal Networks. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_5
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