On the parameterized complexity of computing good edge-labelings

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.

💡 Research Summary

The paper initiates a systematic study of the computational complexity of Good Edge‑Labeling (GEL) and its natural variant c‑GEL, where the number of distinct labels is bounded by a fixed integer c. A good edge‑labeling of a graph G is a real‑valued assignment to the edges such that for every ordered pair of vertices (x, y) there are not two distinct non‑decreasing (i.e., “increasing”) paths from x to y. This concept originates from routing and wavelength‑assignment problems in optical networks.

First, the authors prove that for every c ≥ 2 the decision problem c‑GEL is NP‑complete even on highly restricted inputs. The hardness is shown by a reduction from NAE‑3SAT: they first establish NP‑completeness of 2‑GEL on bounded‑degree bipartite graphs that already admit a 3‑gel, and then lift the result to any c ≥ 2 by a simple gadget that transforms a 2‑gel instance into a c‑gel instance while preserving the property that the graph admits a (c + 1)‑gel. Consequently, limiting the number of labels does not make the problem easier.

Having established intractability, the paper turns to parameterized algorithms. Two structural parameters admit very simple polynomial kernels:

• Neighborhood diversity – a linear‑size kernel is obtained by repeatedly removing vertices that are twins or whose incident edges can be merged without affecting feasibility.
• Vertex cover – a quadratic‑size kernel follows from analogous reduction rules that bound the number of “relevant” edges incident to the cover.

The authors then focus on a more refined structural parameter, the size k of a star‑forest modulator (sfm): a set X of vertices whose removal leaves a disjoint union of stars. This parameter lies between vertex cover and feedback vertex set. They design an FPT algorithm for GEL parameterized by k. After applying the kernel reductions, all remaining stars are classified into three types. Two types are easy to handle; the third type (stars whose centre and all leaves have exactly one neighbour in X) is challenging. To cope with it, the authors introduce labeling relations, which capture the relative order of labels on pairs of edges. By guessing a labeling relation on a set of edges whose size is bounded by a function of k, they reduce each sub‑instance to a 2‑SAT formula. Since 2‑SAT is solvable in polynomial time, the whole algorithm runs in O*(2^{O(k)}) time, establishing that GEL is FPT with respect to sfm.

Next, the paper investigates parameterizations involving treewidth (tw). While c‑GEL can be expressed in monadic second‑order logic (MSOL) with a formula size depending on c, yielding an FPT result via Courcelle’s theorem, the authors provide a concrete dynamic‑programming (DP) algorithm running in c^{tw²}·n time. The DP tables store, for each pair of vertices in a bag, whether certain minimal increasing/decreasing paths exist; this information suffices to decide the existence of a good labeling.

They also consider the combined parameter tw + Δ, where Δ is the maximum degree. Here the number of labels is not bounded a priori. By orienting the line graph partially and tracking “labeling relations” across bags, they obtain an algorithm with running time 2^{O(tw·Δ² + tw²·log Δ)}·n. This DP can either construct a gel that uses the minimum possible number of labels or certify that the input graph is bad.

Finally, the authors answer an open question from Bermond et al. (2013) by proving that deciding whether a graph admits a UPP‑orientation (an orientation where each unordered vertex pair has at most one directed path) is NP‑complete, again via a reduction from NAE‑3SAT.

The paper concludes with several open problems: whether GEL is FPT parameterized solely by feedback vertex set or by treewidth, whether there exists a function f such that any good graph of treewidth tw is f(tw)‑good (i.e., admits a gel with at most f(tw) labels), and analogous questions for the maximum degree. It also asks whether the problem becomes FPT when the modulator is to paths of length at most four, and whether polynomial kernels exist for GEL parameterized by sfm.

Overall, the work provides the first non‑trivial algorithmic results for GEL and c‑GEL, introducing a novel 2‑SAT‑based FPT technique, polynomial kernels for natural structural parameters, and DP algorithms for treewidth‑related parameters. These contributions not only advance the theoretical understanding of good edge‑labelings but also have potential applications in optical network routing and temporal graph design.