I'm interested in NP-hard problems that have a "nice" integer-programming formulation (quadratic or linear, with quadratic or linear constraints) that is not binary.
Of course it is always possible to replace integer-valued variables with a set of binary variables, but that's okay; I'm still interested in optimization problems whose "natural" formulation would be as an integer problem with more-than-binary integers.
Classical examples that come up when searching for integer programming are Set Cover and Max Cut, but those have a very straight-forward binary formulation... any pointers?
EDIT: Let me clarify natural through an example. Let's say we have some integer optimization problem where variables $x_i$ refer to how many units of product $i$ a factory should produce. This would be naturally formulated with integers other than 0 and 1. On the other hand, a knapsack combinatorial optimization problem would have an integer formulation where $x_i = 0, 1$ indicating whether or not item $i$ is chosen.
EDIT: In the integer programming literature, binary does not refer to the overall representation of integers. It refers to the additional constraint that all variables be either 0 or 1, instead of arbitrary integers.