A 3/2-approximation for big two-bar charts packing

Adil I. Erzin, Georgii Melidi, Stepan Nazarenko, Roman V. Plotnikov

Abstract: We consider a 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. Earlier, we proposed an \(O(n^2)\)–time algorithm that constructs the packing of n arbitrary 2-BCs, whose length is at most \(2\cdot OPT+1\), where OPT is the minimum packing length. This paper proposes two new 3/2–approximate algorithms based on sequential matching. One has time complexity \(O(n^4)\) and is applicable when at least one bar of each 2-BC is greater than 1/2. Another has time complexity \(O(n^{3.5})\) and is applicable when, additionally, all BCs are non-increasing or non-decreasing. We prove the estimate’s tightness and conduct a simulation to compare the constructed packings with the optimal solutions or a lower bound of optimum.