Generally 'on the assumption' but 'with the assumption' can work too. 'On the assumption' implies you did something because you assumed something. 'With the assumption' seems slightly less causal/logically connected to me - you did the thing while also believing that thing; it's another factor. 'I booked the restaurant on the assumption that you would be back in time.' 'I booked the restaurant because I know you like pasta, with the assumption that you would be back in time.'