On generic complexity of the subset sum problem for~semigroups of~integer matrices

A. N. Rybalov

Prikladnaâ diskretnaâ matematika, no. 4 (2020), pp. 118-126

Résumé

Generic-case approach to algorithmic problems was suggested by Miasnikov, Kapovich, Schupp, and Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. The subset sum problem is a classic combinatorial problem that has been studied for many decades. Myasnikov, Nikolaev and Ushakov in 2015 introduced an analogue of this problem for arbitrary groups (semigroups). For some classes of groups, such as hyperbolic and nilpotent groups, this problem is solvable in polynomial time. For others, for example, Baumslag — Solitaire groups, group of second order integer unimodular matrices $SL_2(\mathbb{Z})$, this problem is NP-complete. From the works of Gurevich, Kai, Fuchs, Cosen, and Liu, it follows that the subset sum problem for the group $SL_2(\mathbb{Z})$ and for the monoid $SL_2(\mathbb{N})$ is polynomially solvable for almost all inputs. In the paper, we study the generic complexity of the subset sum problem for semigroups of matrices of arbitrary order with integer non-negative elements. This problem is NP-complete, and therefore for it, provided $\text{P} \neq \text{NP} $, there is no polynomial algorithm that solves it for all inputs. We present a polynomial generic algorithm based on the dynamic programming and prove that this problem is generically solvable in polynomial time.

Détail

Keywords: generic complexity, the subset sum problem, integer matrix semigroups.