478 research outputs found
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Exponent equations in HNN-extensions
We consider exponent equations in finitely generated groups. These are
equations, where the variables appear as exponents of group elements and take
values from the natural numbers. Solvability of such (systems of) equations has
been intensively studied for various classes of groups in recent years. In many
cases, it turns out that the set of all solutions on an exponent equation is a
semilinear set that can be constructed effectively. Such groups are called
knapsack semilinear. Examples of knapsack semilinear groups are hyperbolic
groups, virtually special groups, co-context-free groups and free solvable
groups. Moreover, knapsack semilinearity is preserved by many group theoretic
constructions, e.g., finite extensions, graph products, wreath products,
amalgamated free products with finite amalgamated subgroups, and HNN-extensions
with finite associated subgroups. On the other hand, arbitrary HNN-extensions
do not preserve knapsack semilinearity. In this paper, we consider the knapsack
semilinearity of HNN-extensions, where the stable letter acts trivially by
conjugation on the associated subgroup of the base group . We show that
under some additional technical conditions, knapsack semilinearity transfers
from base group to the HNN-extension . These additional technical
conditions are satisfied in many cases, e.g., when is a centralizer in
or is a quasiconvex subgroup of the hyperbolic group .Comment: A short version appeared in Proceedings of ISSAC 202
Knapsack problems in products of groups
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.Comment: 15 pages, 5 figures. Updated to include more general results, mostly
in Section
Compressed Decision Problems in Hyperbolic Groups
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
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