INVERSE PARTITION PROBLEMS

We analyze a partition problem and its inverse problem both in discrete variables and in two continuous ones through dynamic programming. We show that an inverse relation and an envelopping relation hold in each case. It is shown that one optimal solution in discrete partition is expressed by the other through either upper-semi inverse function or lower-semi inverse function and that optimal solutions in continuous partition through the (regular) inverse function. As a result, we show that the optimal partition is to partition equally in essence any quantity into the quantities of the same size of $ e $. We call this optimal policy Euler partition rule