Dag Realizations of Directed Degree Sequences

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.