On the parameterized complexity of computing good edge-labelings

TL;DR

This paper studies the computational complexity of finding good edge-labelings in graphs, introducing parameterized algorithms and proving NP-completeness results for various restricted cases, with applications in optical networks.

Contribution

It provides the first parameterized complexity analysis of the GEL problem, including polynomial kernels and fixed-parameter algorithms based on graph parameters.

Findings

GEL is NP-complete for all c ≥ 2 on restricted graphs.

Polynomial kernels exist for GEL parameterized by neighborhood diversity or vertex cover.

An FPT algorithm for GEL using a 2-SAT formulation based on a modulator to a forest of stars.

Abstract

A good edge-labeling (gel for short) of a graph $G$ is a function $\lambda: E(G) \to \mathbb{R}$ such that, for any ordered pair of vertices $(x, y)$ of $G$, there do not exist two distinct increasing paths from $x$ to $y$, where ``increasing'' means that the sequence of labels is non-decreasing. This notion was introduced by Bermond et al. [Theor. Comput. Sci. 2013] motivated by practical applications arising from routing and wavelength assignment problems in optical networks. Prompted by the lack of algorithmic results about the problem of deciding whether an input graph admits a gel, called GEL, we initiate its study from the viewpoint of parameterized complexity. We first introduce the natural version of GEL where one wants to use at most $c$ distinct labels, which we call $c$-GEL, and we prove that it is NP-complete for every $c \geq 2$ on very restricted instances. We then provide several positive results, starting with simple polynomial kernels for GEL and $c$-\GEL parameterized by neighborhood diversity or vertex cover. As one of our main technical contributions, we present an FPT algorithm for GEL parameterized by the size of a modulator to a forest of stars, based on a novel approach via a 2-SAT formulation which we believe to be of independent interest. We also present FPT algorithms based on dynamic programming for $c$-GEL parameterized by treewidth and $c$, and for GEL parameterized by treewidth and the maximum degree. Finally, we answer positively a question of Bermond et al. [Theor. Comput. Sci. 2013] by proving the NP-completeness of a problem strongly related to GEL, namely that of deciding whether an input graph admits a so-called UPP-orientation.