On the parameterized complexity of computing good edge-labelings

TL;DR

This work systematically analyzes the parameterized complexity of computing good edge-labelings (gel) and its at-most-c labels variant (c-GEL). It proves c-GEL is NP-hard for every c≥2, while also placing GEL and c-GEL in NP, and provides several positive algorithmic results: polynomial kernels for GEL and c-GEL with neighborhood diversity and vertex cover, an FPT algorithm for GEL parameterized by a star-forest modulator, and explicit FPT DP approaches parameterized by treewidth with either c or Δ as the second parameter. A novel 2-SAT formulation is central to handling the sfm-based FPT algorithm, and a line-graph orientation framework underpins the tw+Δ DP. Finally, the NP-hardness of finding a UPP-orientation resolves an open question, highlighting the broader computational difficulty of temporal-orientation problems. Collectively, the results significantly advance the algorithmic understanding of GEL and its relatives, establishing foundational FP T techniques and identifying key open directions for treewidth-based and modulator-based parameterizations.

Abstract

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.

Figures (9)

• Figure 1: A bad graph: for any edge-labeling, in the central 5-cycle there are three adjacent edges $uv,vw,wx$ forming an increasing path $P_1$. But then, there are two other internally-disjoint $(u,x)$-paths $P_2$ and $P_3$ of length two. Given that a 2-path is either increasing or decreasing, two of $P_1,P_2,P_3$ are either increasing or decreasing paths.
• Figure 2: The propagation gadget $P$ along with a 2- gel: blue edges have label one and red edges have label two.
• Figure 5: The extremal gadget $X$.
• Figure 6: The $c$-color gadget $D_c$ for $c=4$ with a $c$- gel where each color represents a label.
• Figure 7: (a) Illustration of the proof of \ref{['claim:0interesting-stars']}, where a $0$-interesting star $S$ in $G \setminus X$ is depicted. The labels of the edges in $E(G) \setminus E(H)$ given by $\lambda'$ are depicted in blue. A cycle $C$ of $G$ containing the center $s$ of $S$ is depicted with thick red edges. (b) Interaction of a $1$-interesting star $S$ with the set $X \cup B$; other stars in $G \setminus X$ are not shown. The arrows in the paths indicate the direction in which the labels increase. Some labels are depicted in blue.

Theorems & Definitions (31)

• Lemma 6: Araújo et al. AraujoCGH12
• Lemma 7: Araújo et al. AraujoCGH12
• Lemma 8: Araújo et al. AraujoCGH12
• Theorem 10
• Lemma 11
• Theorem 12
• Lemma 16
• Lemma 17
• Lemma 18
• Lemma 19