Updated estimates for algorithms for packing $2$-bar charts in a strip

We consider a two-bar charts packing problem in which it is necessary to pack bar charts consisting of two bars in a unit-height strip of minimum length. Each bar has a height of at most $1$ and unit length. The problem under consideration is NP-hard and generalizes the bin packing problem and two-dimensional vector packing problem. This paper proves updated accuracy estimates and time complexity for several previously developed polynomial approximation algorithms for the two-bar charts packing problem and particular cases of the problem. We show the attainability of the estimates. Furthermore, we consider a problem of packing an unlimited number of bar charts belonging to $k$ different types and propose a polynomial algorithm to solve the problem in case $k = \text{const}.$ Illustr. 12, bibliogr. 26.