Complexity of a weighted subset selection problem.
I'm going to assume your problem is equivalent to what this computes:

max map SubsetsOfSize(N) \S (sum map attributes \A (max map S \e A[e]))


In other words, find the best set of items such that picking the best available item attribute values gives the largest sum.

Yes, it's NP-Complete. You can encode 3-Sat into it. Each variable becomes a true-row and false-row. Each clause becomes a column with a 1 in the rows corresponding to its satisfying assignments. You prevent true-row and false-rows from being selected simultaneously by adding columns with huge values in those two rows (0 everywhere else) and requiring that exactly half the rows be selected (thus the optimal solution will require each huge value, and picking a true and false will lose a huge value somewhere else). The 3-sat instance being satisfiable is equivalent to being able to select items for a total score of HugeValue*NumVariables + NumClauses.

Example transformation:

(A or B or C) and (not A or B or not C)

   uA uB uC  C1 C2
+A  9  0  0  1  0
-A  9  0  0  0  1
+B  0  9  0  1  1
-B  0  9  0  0  0
+C  0  0  9  1  0
-C  0  0  9  0  1
N = 3