On the complexity of finding well-balanced orientations with upper bounds on the out-degrees

Abstract

We show that the problem of deciding whether a given graph G has a well-balanced orientation $$\vec {G}$$ G → such that $$d_{\vec {G}}^+(v)\le \ell (v)$$ d G → + ( v ) ≤ ℓ ( v ) for all $$v \in V(G)$$ v ∈ V ( G ) for a given function $$\ell :V(G)\rightarrow \mathbb {Z}_{\ge 0}$$ ℓ : V ( G ) → Z ≥ 0 is NP-complete. We also prove a similar result for best-balanced orientations. This improves a result of Bernáth, Iwata, Király, Király and Szigeti and answers a question of Frank.

Eulerian orientations and vertex-connectivity

“…Further, Bernáth and Joret proved that it is NP-complete to decide if a given graph has a well-balanced orientation respecting a preorientation of some edges [4]. More recently, the author and Szigeti showed that it is NP-complete to decide if a given graph has a wellbalanced orientation respecting only upper bounds on the outdgrees of every vertex [13]. This work will later play a crucial role in the present article.…”

“…The following hardness result immediately follows from the construction in the hardness proof of UBWBO in [13].…”

“…The NP-hardness of UBWBO has been proven by the author and Szigeti [13]. Actually, we need two technical strengthenings of this result where we restrict the instances to satisfy a collection of technical extra conditions.…”

“…The technical definitions of these problems will be postponed to Section 3.2. While the hardness of FSUBWBO is an immediate consequence of the construction on [13], it takes a significant effort to conclude the hardness of SSUBWBO from the hardness of FSUBWBO. After, we conclude the hardness of OCO from the hardness of SSUBWBO.…”