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

An LP with Integrality Gap 1+epsilon for Multidimensional Knapsack