How do minimally violated k-mod cuts work (intuitive explanation)?
Suppose you have an integer linear program, and it contains the following constraints:

x1 + x2 + x3 >= 1,
x1 + x2 + x4 >= 1,
x1 + x3 + x4 >= 1,
x2 + x3 + x4 >= 1.

If we set x1, ..., x4 to 1/3, we satisfy all of the constraints, but we don't have integer values.

Now multiply each constraint by 1/3 and sum them together. You obtain:

x1 + x2 + x3 + x4 >= 4/3.

Since the variables are integer-constrained, we can round up the right-hand side to obtain:

x1 + x2 + x3 + x4 >= 2.

This "cuts off" the above-mentioned fractional solution, so it is called a "cutting plane".

Since we multiplied the constraints by 1/3, the given cutting plane is a "mod-3" cut. Moreover, the fractional solution violates the cutting plane by 2/3, which is best possible. So the mod-3 cut is "maximally violated".

More generally, to obtain a mod-k cut, you multiply each constraint by a multiple of 1/k, before summing and rounding. Such a cut is "maximally violated" by a given fractional solution if it is violated by (k-1)/k.