Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.03447

Going Beyond Surfaces in Diameter Approximation

Michał Włodarczyk
View original source ↗Read source document ↗Export metadata ↓

Abstract

Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known $(1+\varepsilon)$-approximation algorithms with running time $poly(1/\epsilon, \log n) \cdot n$. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving $(1+\varepsilon)$-approximation algorithms with the following running times: 1. $O_h((1/\varepsilon)^{O(h)} \cdot n \log^2 n)$-time algorithm for graphs excluding an apex graph of size h as a minor, 2. $O_d((1/\varepsilon)^{O(d)} \cdot n \log^2 n)$-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+\varepsilon)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time $O_h((1/\varepsilon)^8\cdot n \log n \log W)$ and query time $O((1/\varepsilon)^2 * \log n \log W)$, where $W$ is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Michał Włodarczyk. 2025-07-04. Going Beyond Surfaces in Diameter Approximation. https://arxiv.org/abs/2507.03447

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↗