Irreducible Bin Packing: Complexity, Solvability and Application to the Routing Open Shop

We introduce the following version of an “inefficient” bin packing problem: maximize the number of bins under the restriction that the total content of any two bins is larger than the bin capacity. There is a trivial upper bound on the optimum in terms of the total size of the items. We refer to the decision version of this problem with the number of bins equal to the trivial upper bound as Irreducible Bin Packing. We prove that this problem is NP-complete in an ordinary sense and derive a sufficient condition for its polynomial solvability. The problem has a certain connection to a routing open shop problem which is a generalization of the metric TSP and open shop, known to be NP-hard even for two machines on a 2-node network. So-called job aggregation at some node of a transportation network can be seen as an instance of a bin packing problem. We show that for a two-machine case a positive answer to the Irreducible Bin Packing problem question at some node leads to a linear algorithm of constructing an optimal schedule subject to some restrictions on the location of that node.