A 4/3 OPT+2/3 Approximation for Big Two-Bar Charts Packing Problem

We consider the two-bar charts packing problem generalizing the strongly NP-hard bin packing problem. We prove that the problem remains strongly NP-hard even if each two-bar chart has at least one bar higher than 1/2. If the first (or second) bar of each two-bar chart is higher than 1/2, we show that the O(n 2 )-time greedy algorithm with lexicographic ordering of two-bar charts constructs a packing of length at most OPT+1, where OPT is optimum, and present an O(n 2.5 )-time algorithm that constructs a packing of length at most 4/3 ・ OPT+2/3 in the NP-hard case where each two-bar chart has at least one bar higher than 1/2.