# Is there a name for this unimodilarity-related property?

Consider an arbitrary integer linear program of the form

$\min f(x,y) \\ @ \ Ax + By \leq c\\ x,y \in \mathbb{Z}_+$

If you continuous-relax the integer constraint and still always get integer points as solutions, we talk about the unimodularity property.

But what if you relax the integer constraints and always get integer solutions for $x$, but not necessarily for $y$? Is there a name for this property, and is this type of problem covered anywhere in literature/research?

• I don't understand the problem. What are the inputs? What are the functions $f,g,h,j$? (What are their ranges, and how are they specified? Are they fixed as part of the problem specification, or are they part of the input somehow? Do they have any structure or any properties, e.g., monotonicity, continuity, etc?) What does the "@" mean? What are $G,H,J$? Are they sets? Are they provided as part of the input? – D.W. Aug 13 '15 at 2:17
• It's just an ILP where the variables are divided into two subsets. The $G,H,J$ row is just the constraint set of the ILP... – luegofuego Aug 13 '15 at 8:53