Knapsack problems in products of groups
Elizaveta Frenkel, Andrey Nikolaev, and Alexander Ushakov
TL;DR
This paper explores how free and direct products of groups affect the computational complexity of knapsack-type problems, providing new insights into their behavior in algebraic structures.
Contribution
It demonstrates that free products preserve complexity while direct products can increase it, and extends results to rational subset membership in amalgamated free products.
Findings
Free products preserve knapsack problem complexity.
Direct products may increase problem complexity.
Complexity results for rational subset membership in amalgamated free products.
Abstract
The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.
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