Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.17835

Parameterized complexity of $k$-Coloring in graphs with no long induced paths

Paweł Rzążewski
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the parameterized complexity of (List) $k$-Coloring in $H$-free graphs, when $H$ is a linear forest (i.e., a disjoint union of paths). We show that for any $s \geq 1$, List $k$-Coloring in $(P_3+sP_1)$-free graphs is fixed-parameter tractable (FPT) with respect to the parameter $k$. This extends the result for $s=1$ of Couturier, Golovach, Kratsch, and Paulusma [Journal of Discrete Algorithms, 2012] and answers their open question. Furthermore, we show two hardness results: * $k$-Coloring is W[1]-hard in $2P_2$-free graphs when parameterized by $k$. * $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. Moreover, assuming the ETH, these problems admit no algorithms solving $n$-vertex instances in time $f(k) \cdot n^{o(k)}$ and $f(t) \cdot n^{o(t/\log t)}$, respectively, for any computable function $f$. The former result resolves in a strong form a long-standing open problem, originally posed by Ho\`ang, Kami\'nski, Lozin, Sawada, and Shu [Algorithmica, 2010]. The latter result answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017].

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paweł Rzążewski. 2026-08-18. Parameterized complexity of $k$-Coloring in graphs with no long induced paths. https://arxiv.org/abs/2608.17835

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↗