On Colorful Vertex and Edge Cover Problems

Sayan Bandyapadhyay; Aritra Banik; Sujoy Bhore

Home/Archive/arXiv

arXiv · 2308.15842

On Colorful Vertex and Edge Cover Problems

Sayan Bandyapadhyay·Aritra Banik·Sujoy Bhore

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an $n$-vertex edge-colored graph $G$ with colors from $\{1, \ldots, \omega\}$ and coverage requirements $r_1, r_2, \ldots, r_\omega$, the goal is to find a minimum-sized set of vertices that are incident on at least $r_i$ edges of color $i$, for each $1 \le i \le \omega$, i.e., we need to cover at least $r_i$ edges of color $i$. Colorful Edge Cover is similar to Colorful Vertex Cover, except here we are given a vertex-colored graph and the goal is to cover at least $r_i$ vertices of color $i$, for each $1 \le i \le \omega$, by a minimum-sized set of edges. These problems have several applications in fair covering and hitting of geometric set systems involving points and lines that are divided into multiple groups. Here, fairness ensures that the coverage (resp. hitting) requirement of every group is fully satisfied. We obtain a $(2+\epsilon)$-approximation for the Colorful Vertex Cover problem in time $n^{O(\omega/\epsilon)}$. Thus, for a constant number of colors, the problem admits a $(2+\epsilon)$-approximation in polynomial time. Next, for the Colorful Edge Cover problem, we design an $O(\omega n^3)$ time exact algorithm, via a chain of reductions to a matching problem. For all intermediate problems in this chain of reductions, we design polynomial-time algorithms, which might be of independent interest.

Explore related subjects

cs.DS cs.CG

Keep this discovery

BibTeX RIS

Sayan Bandyapadhyay, Aritra Banik, Sujoy Bhore. 2023-08-30. On Colorful Vertex and Edge Cover Problems. https://arxiv.org/abs/2308.15842

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

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

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↗