Solving the Temporal Knapsack Problem via Recursive Dantzig–Wolfe Reformulation

• We solve the Temporal Knapsack Problem using what we call a Multilevel Dantzig–Wolfe Reformulation (DWR) method.
• The idea of Multilevel DWR is to solve a Mixed Integer Program by recursively applying DWR.
• The method allows fast computation of strong bounds.

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig–Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursively applying DWR, i.e., by using DWR not only for solving the original MIP but also for recursively solving the pricing sub-problems. In a binary case (like the TKP), the Recursive DWR method can be performed in such a way that the only two components needed during the optimization are a Linear Programming solver and an algorithm for solving Knapsack Problems. The Recursive DWR allows us to solve Temporal Knapsack Problem instances through computation of strong dual bounds, which could not be obtained by exploiting the best-known previous approach based on DWR.