Extensor-Coding

TL;DR

This paper introduces a novel exterior algebra-based algorithm for approximately counting paths of length k in directed graphs, with applications to subgraph isomorphism and polynomial multilinear term detection.

Contribution

It presents a new exterior algebra approach for path counting, generalizes to subgraph isomorphism with bounded pathwidth, and offers a randomized method for detecting multilinear polynomial terms.

Findings

Approximate path counting in time   \, ext{error}

Deterministic algorithm for k-path detection in graphs with few paths

Randomized detection of multilinear terms in algebraic circuits

Abstract

We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic $2^{k}\cdot\operatorname{poly}(n)$ time algorithm to find a $k$-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect $k$-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.