Abstract
Funnels are a new natural subclass of DAGs. Intuitively, a DAG is a funnel if every source-sink path can be uniquely identified by one of its arcs. Funnels are an analog to trees for directed graphs that is more restrictive than DAGs but more expressive than in-/out-trees. Computational problems such as finding vertex-disjoint paths or tracking the origin of memes remain NP-hard on DAGs while on funnels they become solvable in polynomial time. Our main focus is the algorithmic complexity of finding out how funnel-like a given DAG is. To this end, we study the NP-hard problem of computing the arc-deletion distance to a funnel of a given DAG. We develop efficient exact and approximation algorithms for the problem and test them on synthetic random graphs and real-world graphs.
M. G. Millani—Partially supported by the DFG, project FPTinP (NI 369/16).
H. Molter—Partially supported by the DFG, project MATE (NI 369/17).
M. Sorge—Supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11 and Israel Science Foundation (grant no. 551145/14).
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Notes
- 1.
There is also a simple \({\mathcal {O}(5^d\cdot \left| V\right| \cdot \left| A\right| )}\)-time algorithm for general digraphs [15].
- 2.
A full version is available on arXiv [17].
- 3.
A graph H is called a topological minor of a graph G if a subgraph of G can be obtained from H by subdividing edges (that is, replacing arcs by directed paths).
- 4.
Listed at https://www.archlinux.org/packages/ and obtained using pacman.
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Millani, M.G., Molter, H., Niedermeier, R., Sorge, M. (2018). Efficient Algorithms for Measuring the Funnel-Likeness of DAGs. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_16
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