algorithms – matrix partition in k subsets

I'm assuming $ sum (i) $ you mean that given an order from the $ k $ partitioned subsets, sum over all the elements of the $ i $the subset.

Here are some thoughts: the $ k = 2 $ case is the optimization variant of the set partition problem (, which is known to be as difficult as the sum of subsets. It seems that the general case may be so difficult, but there does not seem to be an easy way to recursively reduce the general case to $ k = 2 $.