The computational complexity of the local postage stamp problem

The well-studied local postage stamp problem (LPSP) is the following: given a positive integer k, a set of postive integers 1 = a1 < a2 < … < ak and an integer h >= 1, what is the smallest positive integer which cannot be represented as a linear combination x1 a1 + … + xk ak where x1 + … + xk <= h and each xi is a non-negative integer? In this note we prove that LPSP is NP-hard under Turing reductions, but can be solved in polynomial time if k is fixed.