Counting Paths and Trees via Exterior Algebra

2026-09-13

AI Summary

本文通过结合外代数与随机矩阵估计器，提供了计算给定图中k-路径和k-森林数量的近似算法，解决了相关领域的开放问题。

Abstract

We give randomized approximation algorithms for counting k-paths and k-forests in a host graph. Here k denotes the number of pattern vertices, n and m denote the numbers of host vertices and edges or arcs, ε is the relative error, and δ is the failure probability. Our main results are: 1. Paths: We approximate the number of directed paths on $k$ vertices in $2^k k^{O(1)}(n+m)\varepsilon^{-2}\log(2/δ)$ arithmetic operations. 2. Trees and forests: For every fixed $η>0$, we approximate the number of non-induced copies of a given forest on $k$ vertices in $(2+η)^k n^{O_η(1)}\varepsilon^{-2}\log(2/δ)$ arithmetic operations. Our path algorithm resolves a conjecture of Koutis and Williams~[CACM 2016] and answers an open question of Lokshtanov, Saurabh, and Zehavi~[SODA 2021] by giving a $2^k poly(n,\varepsilon^{-1})$-time approximation scheme. Our algorithms combine exterior algebra with random matrix estimators, using the tensor-train moment bound of Rakhshan and Rabusseau~[AISTATS 2020]. For forests, we use a small-component separator to evaluate the estimator efficiently.

Problem

• counting
• paths
• forests
• approximation
• algorithms

Innovation

• Exterior Algebra
• Randomized Approximation Algorithms
• Tensor-Train Moment Bound