Abstract
We consider the following graph realization problem. Given a sequence \(S:={a_1 \choose b_1},\dots,{a_n \choose b_n}\) with \(a_i,b_i\in \mathbb{Z}_0^+\), does there exist an acyclic digraph (a dag, no parallel arcs allowed) G = (V,A) with labeled vertex set V: = {v 1,…,v n } such that for all v i ∈ V indegree and outdegree of v i match exactly the given numbers a i and b i , respectively? The complexity status of this problem is open, while a generalization, the f-factor dag problem can be shown to be NP-complete. In this paper, we prove that an important class of sequences, the so-called opposed sequences, admit an O(n + m) realization algorithm, where n and \(m = \sum_{i=1}^n a_i = \sum_{i=1}^n b_i\) denote the number of vertices and arcs, respectively. For an opposed sequence it is possible to order all non-source and non-sink tuples such that a i ≤ a i + 1 and b i ≥ b i + 1. Our second contribution is a realization algorithm for general sequences which significantly improves upon a naive exponential-time algorithm. We also investigate a special and fast realization strategy “lexmax”, which fails in general, but succeeds in more than 97% of all sequences with 9 tuples.
This work was supported by the DFG Focus Program Algorithm Engineering, grant MU 1482/4-2.
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Berger, A., Müller-Hannemann, M. (2011). Dag Realizations of Directed Degree Sequences. In: Owe, O., Steffen, M., Telle, J.A. (eds) Fundamentals of Computation Theory. FCT 2011. Lecture Notes in Computer Science, vol 6914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22953-4_23
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DOI: https://doi.org/10.1007/978-3-642-22953-4_23
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