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Maximum Reachability Orientation of Mixed Graphs

Author Florian Hörsch

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LIPIcs.STACS.2026.53.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Network optimization
  • Theory of computation → Approximation algorithms analysis
Keywords
  • orientations
  • mixed graphs
  • reachability
  • parameterized complexity
  • approximation

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Abstract

We aim to find orientations of mixed graphs optimizing the total reachability, a problem that has applications in causality and biology. For given a digraph D, we use P(D) for the set of ordered pairs of distinct vertices in V(D) and we define κ_D:P(D) → {0,1} by κ_D(u,v) = 1 if v is reachable from u in D, and κ_D(u,v) = 0, otherwise. We use R(D) = ∑_{(u,v) ∈ P(D)}κ_D(u,v).
Now, given a mixed graph G, we aim to find an orientation x⃑{G} of G that maximizes R(x⃑{G}). Hakimi, Schmeichel, and Young proved that the problem can be solved in polynomial time when restricted to undirected inputs. They inquired about the complexity in mixed graphs.
We answer this question by showing that this problem is NP-hard, and, moreover, APX-hard.
We then develop a finer understanding of how quickly the problem becomes difficult when going from undirected to mixed graphs. To this end, we consider the parameterized complexity of the problem with respect to the number k of preoriented arcs of G, a poorly studied form of parameterization.
We show that the problem can be solved in time n^{O(k)} and that a (1-ε)-approximation can be computed in time f(k,ε)n^{O(1)} for any ε > 0.

Cite As Get BibTex

Florian Hörsch. Maximum Reachability Orientation of Mixed Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.53

Author Details

Florian Hörsch
  • CISPA, Saarbrücken, Germany

Acknowledgements

I want to thank Charupriya Sharma for making me aware of the connection to causality.

References

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