Related papers: Knapsack problem for automaton 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…

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In previous work, the authors proved that for each graph group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine the exact…

The multiple knapsack problem with grouped items aims to maximize rewards by assigning groups of items among multiple knapsacks, considering knapsack capacities. Either all items in a group are assigned or none at all. We propose algorithms…

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for…

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge…

We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form…

The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However we prove that, in the most general case, the problem is undecidable. We study the case of automaton…

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or…

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the…

The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the…

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under…

This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating step-off, a…

Computing sets of high quality solutions has gained increasing interest in recent years. In this paper, we investigate how to obtain sets of optimal solutions for the classical knapsack problem. We present an algorithm to count exactly the…

We study the knapsack problem with group fairness constraints. The input of the problem consists of a knapsack of bounded capacity and a set of items, each item belongs to a particular category and has and associated weight and value. The…

(Free-abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, orbit undecidable, free subgroups of GL_d(Z), for d > 5, and…

We prove that Knapsack problem (KP) is undecidable for any group of nilpotency class two if the number of generators (without torsion) of the derived subgroup is at least 322. This result together with the fact that if KP is undecidable for…

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton A, are there words accepted by A with probability arbitrarily close to 1? This problem was proved undecidable recently.…

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy…

This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem…