Abstract
We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting.
Highlights
The study of discrete optimization problems in groups was initiated in [10], where the authors introduced group-theoretic generalizations of the classical knapsack problem and its variations, e.g., the subset sum problem and bounded submonoid membership problem.In the subsequent papers [13] and [14], the authors studied generalizations of the Post corresponce problem and classical lattice problems in groups
The Post correspondence problem in G is closely related to the twisted conjugacy problem in G, the equalizer problem in G, and a strong version of the word problem
Lattice problems are related to the classical subgroup membership problem and finite state automata
Summary
The study of discrete optimization problems in groups was initiated in [10], where the authors introduced group-theoretic generalizations of the classical knapsack problem and its variations, e.g., the subset sum problem and bounded submonoid membership problem. In the subsequent papers [13] and [14], the authors studied generalizations of the Post corresponce problem and classical lattice problems in groups. Lattice problems are related to the classical subgroup membership problem and finite state automata. We prove NP-hardness of the subset sum problem in any finitely generated weakly regular branch group. For groups with polynomial time word problem, e.g., the first Key words and phrases: Grigorchuk group, branch groups, subset sum problem, NP-completeness
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
