Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P

TL;DR

This paper advances the theory of power circuits and presents new algorithms that solve the Word Problem in Higman's group in polynomial time, significantly improving previous complexity bounds.

Contribution

It introduces a quadratic time reduction procedure for power circuits and applies it to solve the Word Problem in Higman's group in polynomial time, a major theoretical breakthrough.

Findings

Power circuits can be optimized with quadratic reduction procedures.

The Word Problem for the Baumslag group is solvable in cubic time.

Higman's group Word Problem is decidable in polynomial time.

Abstract

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound has been non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. The improvement is crucial for our other results. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.