A Finitely presented group whose word problem has sampleable hard instances

TL;DR

This paper presents a finitely presented group with a word problem that has a complexity core which can be sampled efficiently, making hard instances easy to find.

Contribution

It introduces a natural decision problem with a sampleable subset of hard instances, bridging the gap between theoretical hardness and practical instance generation.

Findings

Hard instances are easy to find due to a sampleable complexity core.

The word problem for the presented group has a linear-time sampleable core.

This demonstrates a natural problem with accessible hard instances.

Abstract

Hard instances of natural computational problems are often elusive. In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find. More precisely the problem has a complexity core sampleable in linear time.