# Difficulty of Integer Linear Programming vs. Mixed Integer Linear Programming

I have recently been working on an applied project where I have to solve optimization problems.

I have found that it is much easier to solve an integer linear program (ILP) as opposed to a mixed-integer linear programs (MILP), in the sense it is faster for me to find feasible solutions as well as to prove optimality.

I am wondering if there is some kind of formal result that confirms this (i.e. on the time complexity of ILP vs. MILP).

To be clear, when I am refering to an ILP I mean an optimization problem of the form

\begin{align} \min_{x} \quad & c^Tx \\ \text{s.t.} \quad & Ax = b \\ \; & x\in\mathbb{Z}^N\end{align}

where the variables are integer-valued so that $x\in\mathbb{Z}^N$ but the parameters are real valued so that $c \in \mathbb{R}^N$, $A \in \mathbb{R}^{M \times N}$, $b \in \mathbb{R}^M$. In turn, an MILP would be different in that at least one of the variables would take on a real value.

• Computational complexity probably won't help you here: Both problems are NP-complete. – DCTLib Sep 15 '14 at 10:15