Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.12363

The Parameterized Complexity of Vertex-Coloring Edge-Weighting

Shubhada Aute·Fahad Panolan·Geevarghese Philip
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by the landmark resolution of the 1-2-3 Conjecture, we initiate the study of the parameterized complexity of the Vertex-Coloring {0,1}-Edge-Weighting problem and its generalization, Vertex-Coloring Pre-edge-Weighting, under various structural parameters. The base problem, Vertex-Coloring {0,1}-Edge-Weighting, asks whether we can assign a weight from {0,1} to each edge of a graph. The goal is to ensure that for every pair of adjacent vertices, the sums of their incident edge weights are distinct. In the Vertex-Coloring Pre-edge-Weighting variant, we are given a graph where a subset of edges is already assigned fixed weights from {0,1}. The goal is to determine if this partial weighting can be extended to all remaining edges such that the final, complete assignment satisfies the proper vertex coloring property. While the existence of such weightings is well-understood for specific graph classes, their algorithmic complexity under structural parameterization has remained unexplored. We prove both hardness and tractability for the problem, across a hierarchy of structural parameters. We show that both the base problem and the Pre-edge-Weighting variant are W[1]-hard when parameterized by the size of a feedback vertex set of the input graph. On the positive side, we establish that the base problem and a restricted Pre-edge-Weighting variant where the pre-assigned weights are all 1, become FPT when parameterized by the size of a vertex cover of the input graph. Further, we show that both the base problem and the Pre-edge-Weighting variant have XP algorithms when parameterized by the treewidth of the input graph.

Explore related subjects

cs.DScs.CCcs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shubhada Aute, Fahad Panolan, Geevarghese Philip. 2026-04-14. The Parameterized Complexity of Vertex-Coloring Edge-Weighting. https://arxiv.org/abs/2604.12363

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗