Did Fourier Really Meet M\"obius? Fast Subset Convolution via FFT

TL;DR

This paper introduces a simple FFT-based algorithm for subset convolution that maintains the original time complexity while avoiding large intermediate outputs and precision errors, making it more practical.

Contribution

The authors present a novel FFT-based method for subset convolution that eliminates the need for set function transforms, improving practical usability without sacrificing efficiency.

Findings

The new algorithm matches the original $O(2^n n^2)$ running time.

It avoids large intermediate outputs and floating-point errors.

Enables practical application of subset convolution techniques.

Abstract

In their seminal work on subset convolution, Bj\"orklund, Husfeldt, Kaski and Koivisto introduced the now well-known $O(2^n n^2)$-time evaluation of the subset convolution in the sum-product ring. This sparked a wave of remarkable results for fundamental problems, such as the minimum Steiner tree and the chromatic number. However, in spite of its theoretical improvement, large intermediate outputs and floating-point precision errors due to alternating addition and subtraction in its set function transforms make the algorithm unusable in practice. We provide a simple FFT-based algorithm that completely eliminates the need for set function transforms and maintains the running time of the original algorithm. This makes it possible to take advantage of nearly sixty years of research on efficient FFT implementations.