Counting on Nowhere Dense Classes

2026-09-16

AI Summary

本文针对稀疏类关系结构，提出了一种算法，通过近线性时间预处理实现一阶公式的计数问题的常数时间求解。

Abstract

For every effectively nowhere dense class $\mathcal{C}$ of relational structures, we present an algorithm that runs an almost-linear-time preprocessing step on a given structure $\mathcal{A} \in \mathcal{C}$ and a first-order formula $φ(x_1, \dots, x_k, y_1, \dots, y_\ell)$. After the preprocessing, whenever given a tuple $\bar{v} \in A^k$, the algorithm computes the number of tuples $\bar{w} \in A^\ell$ that satisfy $\mathcal{A} \models φ(\bar{v}, \bar{w})$ in constant time. Building on this, we provide an algorithm for constant-time query answering and constant-delay enumeration after almost-linear-time preprocessing for the recently introduced logic clique-guarded first-order logic with counting (cgFOC) on effectively nowhere dense classes. This generalises the testing and enumeration results for first-order logic [Schweikardt, Segoufin, and Vigny, JACM 2022] and the evaluation result for the first-order logic with counting FOC1 [Grohe and Schweikardt, PODS 2018] on nowhere dense classes.

Problem

• nowhere dense classes
• first-order logic
• counting
• preprocessing
• query answering

Innovation

• nowhere dense classes
• constant-time query answering
• clique-guarded first-order logic with counting (cgFOC)