Weighted proper orientations of trees and graphs of bounded treewidth

Given a simple graph $G$, a weight function $w:E(G)\rightarrow \mathbb{N} \setminus \{0\}$, and an orientation $D$ of $G$, we define $\mu^-(D) = \max_{v \in V(G)} w_D^-(v)$, where $w^-_D(v) = \sum_{u\in N_D^{-}(v)}w(uv)$. We say that $D$ is a weighted proper orientation of $G$ if $w^-_D(u) \neq w^-_D(v)$ whenever $u$ and $v$ are adjacent. We introduce the parameter weighted proper orientation number of $G$, denoted by $\overrightarrow{\chi}(G,w)$, which is the minimum, over all weighted proper orientations $D$ of $G$, of $\mu^-(D)$. When all the weights are equal to 1, this parameter is equal to the proper orientation number of $G$, which has been object of recent studies and whose determination is NP-hard in general, but polynomial-time solvable on trees. Here, we prove that the equivalent decision problem of the weighted proper orientation number (i.e., $\overrightarrow{\chi}(G,w) \leq k$?) is (weakly) NP-complete on trees but can be solved by a pseudo-polynomial time algorithm whose running time depends on $k$. Furthermore, we present a dynamic programming algorithm to determine whether a general graph $G$ on $n$ vertices and treewidth at most ${\sf tw}$ satisfies $\overrightarrow{\chi}(G,w) \leq k$, running in time $O(2^{{\sf tw}^2}\cdot k^{3{\sf tw}}\cdot {\sf tw} \cdot n)$, and we complement this result by showing that the problem is W[1]-hard on general graphs parameterized by the treewidth of $G$, even if the weights are polynomial in $n$.