An LP with Integrality Gap 1+epsilon for Multidimensional Knapsack

In this note we study packing or covering integer programs with at most k constraints, which are also known as k-dimensional knapsack problems. For integer k > 0 and real ǫ > 0, we observe there is a polynomial-sized LP for the k-dimensional knapsack problem with integrality gap at most 1+ ǫ. The variables may be unbounded or have arbitrary upper bounds. In the (classical) packing case, we can also remove the dependence of the LP on the cost-function, yielding a polyhedral approximation of the integer hull. This generalizes a recent result of Bienstock [3] on the classical knapsack problem.