Formulations and algorithms for the recoverable $${\varGamma }$$ Γ -robust knapsack problem

One of the most frequently occurring substructures in integer linear programs (ILPs) is the knapsack constraint. In this paper, we study ways to deal with uncertainty in the coefficients of such constraints. We combine the budget uncertainty set of Bertsimas and Sim (Math Program Ser B 98:49–71, 2003; Oper Res 52(1):35–53, 2004) with a recovery action, i.e., in order to restore feasibility up to k items may be removed when the actual coefficients are known. We present three different approaches to formulate this recoverable robust knapsack (rrKP) as ILP, including a novel compact reformulation of quadratic size. The other two formulations have exponentially many variables and/or constraints. To keep the ILPs small in practice, we develop separation algorithms, not only for the exponential formulations, but also for the compact reformulation. An experimental comparison of six different approaches to solve the rrKP on a carefully designed set of benchmark instances reveals that a lazy constraint-and-variables approach for the compact reformulation outperforms other alternatives.