Deciding multiple tiling by polygons in polynomial time

Suppose P is a symmetric convex polygon in the plane. We give an algorithm, running in polynomial time in the number of sides of the polygon, which decides if P can tile the plane by translations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each j=1,2,… ,n one of two given vectors e_j or τ _j so that the selection spans a discrete additive subgroup.