Weighted proper orientations of trees and graphs of bounded treewidth

TL;DR

This paper introduces the weighted proper orientation number for graphs, proves its computational complexity on trees, and provides algorithms for graphs with bounded treewidth, highlighting both hardness and tractability results.

Contribution

It defines the weighted proper orientation number, proves NP-completeness on trees, and develops a dynamic programming algorithm for graphs with bounded treewidth.

Findings

NP-complete on trees for weighted proper orientation number

Pseudo-polynomial time algorithm for trees

Fixed-parameter tractable algorithm for graphs with bounded treewidth

Abstract

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 $n$.