An Improved Approximation for Packing Big Two-Bar Charts

TL;DR

This paper introduces improved approximation algorithms for the Two-Bar Charts Packing Problem, achieving better ratios and efficiency for specific cases where bars have heights above 1/2.

Contribution

It presents new approximation algorithms with improved ratios and computational complexity for packing 2-BCs under certain height constraints.

Findings

Achieved a 16/11 approximation ratio with O(n^3) time for cases with bars at least 1/2 in height.

Developed a 5/4 approximation algorithm with O(n^{2.5}) time for non-increasing or non-decreasing 2-BCs.

Provided algorithms that outperform previous methods for specific 2-BC packing scenarios.

Abstract

Recently, we presented a new Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem and 2-D Vector Packing Problem. Earlier, we have proposed several polynomial approximation algorithms. In particular, when each 2-BC has at least one bar of height more than 1/2, we have proposed a 3/2--approximation polynomial algorithm. This paper proposes an $O(n^3)$--time 16/11--approximation algorithm for packing 2-BCs when at least one bar of each BC has a height not less than 1/2 and an $O(n^{2.5})$--time 5/4--approximation algorithm for packing non-increasing or non-decreasing 2-BCs when each 2-BC has at least one bar which height is more than 1/2, where $n$ is the number of 2-BCs.