The sacrifice precision method

Sometimes problems are only hard when the numerical quantities are arbitrary, complicated large integers. When these numerical quantities are limited to being small integers, the problems become efficiently solvable. (One way to formalize this is to consider these quantities as being specified in unary, and ask if the problem becomes polynomial-time solveable.) This is called being “hard in the weak sense”. In this case, it frequently makes sense to round the input parameters to small integers, and hope (prove) that the solution to this “approximated instance” is an approximate solution to the actual instance. We can frequently view this “sacrifice precision” technique as a special kind of reduction, where a general instance of the problem is reduced to a “small integer” instance.