[cperciva]

look for a short(est) vector of this lattice

Just to elaborate slightly on this: Finding the shortest vector in a lattice is (in general) hard, but finding a vector which is no more than a constant times the length of the shortest vector is easy. So techniques which rely on lattice reduction usually construct a lattice such that the shortest vector is much shorter than the second-shortest vector -- in other words, the shortest vector is the only one which satisfies the "no more than X times the length of the shortest vector" condition.

In the case of integer relation finding, this translates into having enough digits and making the value K large enough.