Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2411.17948

Structural Parameterization of Locating-Dominating Set and Test Cover

Dipayan Chakraborty·Florent Foucaud·Diptapriyo Majumdar·Prafullkumar Tale
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if there is a subset $S$ of $k$ vertices such that any two distinct vertices not in $S$ are dominated by distinct subsets of $S$. In the second problem, an input is a set of items $U$, a set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, Prafullkumar Tale. 2024-11-26. Structural Parameterization of Locating-Dominating Set and Test Cover. https://arxiv.org/abs/2411.17948

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↗