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Approximately Counting Approximately-Shortest Paths in Directed Acyclic Graphs

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Abstract

Given a directed acyclic graph with non-negative edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. This is best possible, as we also show that the problem is #P-complete. We then show that, unless P=NP, there is no finite approximation to the bi-criteria version of the problem: count the number of s-t paths of length at most L 1 in the first criterion, and of length at most L 2 in the second criterion. On the positive side, we extend the approximation scheme for the relaxed version of the problem, where, given thresholds L 1 and L 2, we relax the requirement of the s-t paths to have length exactly at most L 1, and allow the paths to have length at most L 1′ : = (1+δ)L 1, for any δ > 0.

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Notes

  1. Note that due to the lack of cycles, the problems of looking for shortest and longest paths on DAGs are computationally identical.

  2. To see this, observe that in a topologically sorted graph G, any subset of V∖{s,t} gives a unique candidate for an s-t path.

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Acknowledgments

We thank Octavian Ganea and anonymous reviewers for their suggestions and comments. The work has been partially supported by the Swiss National Science Foundation under grant no. 200021_138117/1, and by the EU FP7/2007-2013, under the grant agreement no. 288094 (project eCOMPASS).

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, Department of Computer Science, ETH Zurich, Switzerland

    Matúš Mihalák, Rastislav Šrámek & Peter Widmayer

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  1. Matúš Mihalák
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  2. Rastislav Šrámek
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  3. Peter Widmayer
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Correspondence to Rastislav Šrámek.

Additional information

Preliminary version of this paper appeared at the 11th Workshop on Approximation and Online Algorithms (WAOA 2013).

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Mihalák, M., Šrámek, R. & Widmayer, P. Approximately Counting Approximately-Shortest Paths in Directed Acyclic Graphs. Theory Comput Syst 58, 45–59 (2016). https://doi.org/10.1007/s00224-014-9571-7

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