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arXiv · 2609.14682

Counting Paths and Trees via Exterior Algebra

Fahad Panolan·Saket Saurabh·Meirav Zehavi·Jie Xue
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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.

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Fahad Panolan, Saket Saurabh, Meirav Zehavi, Jie Xue. 2026-09-13. Counting Paths and Trees via Exterior Algebra. https://arxiv.org/abs/2609.14682

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