  We study a family of combinatorial optimization problems defined by a
parameter $p\in[0,1]$, which involves spectral functions applied to positive
semidefinite matrices, and has some application in the theory of optimal
experimental design. This family of problems tends to a generalization of the
classical maximum coverage problem as $p$ goes to 0, and to a trivial instance
of the knapsack problem as $p$ goes to 1.
  In this article, we establish a matrix inequality which shows that the
objective function is submodular for all $p\in[0,1]$, from which it follows
that the greedy approach, which has often been used for this problem, always
gives a design within $1-1/e$ of the optimum. We next study the design found by
rounding the solution of the continuous relaxed problem, an approach which has
been applied by several authors. We prove an inequality which generalizes a
classical result from the theory of optimal designs, and allows us to give a
rounding procedure with an approximation factor which tends to 1 as $p$ goes to
1.
