Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.15080

Minimum Sum Coloring with Bundles in Trees and Bipartite Graphs

Takehiro Ito·Naonori Kakimura·Naoyuki Kamiyama·Yusuke Kobayashi·Yoshio Okamoto
View original source ↗Read source document ↗Export metadata ↓

Abstract

The minimum sum coloring problem with bundles was introduced by Darbouy and Friggstad (SWAT 2024) as a common generalization of the minimum coloring problem and the minimum sum coloring problem. During their presentation, the following open problem was raised: whether the minimum sum coloring problem with bundles could be solved in polynomial time for trees. We answer their question in the negative by proving that the minimum sum coloring problem with bundles is NP-hard even for paths. We complement this hardness by providing algorithms of the following types. First, we provide a fixed-parameter algorithm for trees when the number of bundles is a parameter; this can be extended to graphs of bounded treewidth. Second, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and the difference between the number of vertices and the number of bundles is constant. Third, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and each bundle induces a connected subgraph. We further show that for bipartite graphs, the problem with weights is NP-hard even when the number of bundles is at least three, but is polynomial-time solvable when the number of bundles is at most two. The threshold shifts to three versus four for the problem without weights.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Yoshio Okamoto. 2025-09-18. Minimum Sum Coloring with Bundles in Trees and Bipartite Graphs. https://arxiv.org/abs/2509.15080

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↗