  We study the proximity of the optimal value of the m-dimensional knapsack
problem to the optimal value of that problem with the additional restriction
that only one type of items is allowed to include in the solution. We derive
exact and asymptotic formulas for the precision of such approximation, i.e. for
the infinum of the ratio of the optimal value for the objective functions of
the problem with the cardinality constraint and without it. In particular, we
prove that the precision tends to 0.59136.../m if n tends to infinity and m is
fixed. Also, we give the class of the worst multi-dimensional knapsack problems
for which the bound is attained. Previously, similar results were known only
for the case m=1.
