Fast Witness Counting

TL;DR

This paper presents an optimal algorithm for the witness-counting problem over finite fields, using Walsh-Hadamard transforms and inclusion-exclusion corrections, with matching lower bounds based on complexity assumptions.

Contribution

It introduces a fine-grained optimal algorithm for witness counting over _2^d, combining Walsh-Hadamard transforms with novel correction techniques and provides matching lower bounds.

Findings

Algorithm runs in ^*(2^d) time with logarithmic dependence on |V|

Introduces correction terms based on equivalence relations to address overcounting

Establishes lower bounds based on  ETH and kernelization impossibility

Abstract

We study the witness-counting problem: given a set of vectors $V$ in the $d$-dimensional vector space over $\mathbb{F}_2$, a target vector $t$, and an integer $k$, count all ways to sum-up exactly $k$ different vectors from $V$ to reach $t$. The problem is well-known in coding theory and received considerable attention in complexity theory. Recently, it appeared in the context of hardware monitoring. Our contribution is an algorithm for witness counting that is optimal in the sense of fine-grained complexity. It runs in time $\mathcal{O}^*(2^d)$ with only a logarithmic dependence on $m=|V|$. The algorithm makes use of the Walsh-Hadamard transform to compute convolutions over $\mathbb{F}_2^d$. The transform, however, overcounts the solutions. Inspired by the inclusion-exclusion principle, we introduce correction terms. The correction leads to a recurrence that we show how to solve efficiently. The correction terms are obtained from equivalence relations over $\mathbb{F}_2^d$. We complement our upper bound with two lower bounds on the problem. The first relies on $\# ETH$ and prohibits an $2^{o(d)}$-time algorithm. The second bound states the non-existence of a polynomial kernel for the decision version of the problem.