[Paper Review] Compressed Decision Problems in Hyperbolic Groups

Derek F. Holt, Markus Lohrey+1 more|arXiv (Cornell University)|Aug 21, 2018

Geometric and Algebraic Topology51 references4 citations

TL;DR

This paper establishes that the compressed word, conjugacy, simultaneous conjugacy, centralizer, and knapsack problems are solvable in polynomial time for hyperbolic groups. The key contribution is a polynomial-time algorithm to compute shortlex-reduced straight-line programs, enabling efficient solutions to multiple decision problems in this class of groups.

ABSTRACT

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight-line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group G, the compressed knapsack problem in G is NP-complete.

Motivation & Objective

• To establish polynomial-time solvability of the compressed word problem in hyperbolic groups.
• To extend this result to compressed versions of the conjugacy, simultaneous conjugacy, centralizer, and knapsack problems.
• To prove that the compressed knapsack problem is NP-complete for infinite hyperbolic groups.
• To demonstrate that the word problem for automorphism and outer automorphism groups of hyperbolic groups is solvable in polynomial time.
• To provide a foundational algorithm for shortlex reduction of group elements represented by straight-line programs in hyperbolic groups.

Proposed method

• Develops a polynomial-time algorithm to compute a shortlex-reduced straight-line program for any given input program in a hyperbolic group.
• Uses the geometric properties of hyperbolic groups, including linear isoperimetric inequalities and quasiconvexity, to bound word lengths and enable efficient reduction.
• Applies results from Gromov hyperbolicity and automatic structures to ensure that group elements of bounded length can be processed in constant time.
• Employs straight-line program (SLP) techniques to represent and manipulate group elements compactly, enabling efficient computation of products and conjugations.
• Reduces the compressed simultaneous conjugacy and centralizer problems to bounded-length word problems via conjugation and subgroup analysis.
• Leverages Theorem 6.6 (bounded solution size for knapsack expressions) to reduce the search space to exponential size in input length, enabling NP verification via SLP evaluation.

Experimental results

Research questions

• RQ1 Can the compressed word problem be solved in polynomial time for hyperbolic groups?
• RQ2 Is the compressed simultaneous conjugacy problem solvable in polynomial time for hyperbolic groups?
• RQ3 Can the compressed centralizer problem be solved efficiently in hyperbolic groups?
• RQ4 What is the computational complexity of the compressed knapsack problem in infinite hyperbolic groups?
• RQ5 Does the solvability of the compressed word problem imply polynomial-time solvability of the word problem for Aut(G) and Out(G) in hyperbolic groups?

Key findings

• The compressed word problem for any hyperbolic group is solvable in polynomial time via a shortlex reduction algorithm for straight-line programs.
• The compressed simultaneous conjugacy problem is solvable in polynomial time, and a conjugating element can be computed in polynomial time when solutions exist.
• The compressed centralizer problem is solvable in polynomial time, with generators of the centralizer computable in constant time for bounded-length inputs.
• For any infinite hyperbolic group, the compressed knapsack problem is NP-complete, with solutions bounded by a polynomial in the input size.
• The word problem for the automorphism and outer automorphism groups of a hyperbolic group is solvable in polynomial time via reductions to the compressed word and simultaneous conjugacy problems.
• The algorithm for shortlex reduction runs in polynomial time and is based on geometric and combinatorial properties of hyperbolic groups, including quasiconvexity and bounded torsion.