A good edge-labeling (gel for short) of a graph $G$ is a function $λ: E(G) \to \mathbb{R}$ 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.

翻译：图的良好边标记（简称gel）是一个函数 $λ: E(G) \to \mathbb{R}$，使得对于 $G$ 中任意有序顶点对 $(x, y)$，不存在两条从 $x$ 到 $y$ 的不同递增路径，其中“递增”意指标签序列非递减。该概念由Bermond等人 [Theor. Comput. Sci. 2013] 提出，其动机源于光网络中路由与波长分配问题的实际应用。由于判定输入图是否存在gel的问题（称为GEL）缺乏算法结果，我们首次从参数化复杂性的角度对其展开研究。首先，我们引入GEL的自然版本，即要求使用至多 $c$ 种不同标签的问题，记作 $c$-GEL，并证明对于任意 $c \geq 2$，即使在高度受限的实例上该问题也是NP完全的。随后，我们提供若干正面结果：首先给出针对GEL和$c$-GEL的简单多项式核，其参数化为邻域多样性或顶点覆盖。作为主要技术贡献之一，我们提出一种基于2-SAT公式化的新颖方法，用于参数化为星林调节器大小的GEL的FPT算法——我们相信该方法本身具有独立的研究价值。此外，我们基于动态规划提出针对$c$-GEL（参数化为树宽和$c$）以及GEL（参数化为树宽和最大度数）的FPT算法。最后，我们通过证明与GEL密切相关的判定问题（即输入图是否允许所谓的UPP定向）的NP完全性，正面回应了Bermond等人 [Theor. Comput. Sci. 2013] 提出的一个开放问题。