  Suppose we are given integer $k \leq n$ and $n$ boxes labeled $1,\ldots, n$
by an adversary, each containing a number chosen from an unknown distribution.
We have to choose an order to sequentially open these boxes, and each time we
open the next box in this order, we learn its number. If we reject a number in
a box, the box cannot be recalled. Our goal is to accept the $k$ largest of
these numbers, without necessarily opening all boxes. This is the free order
multiple-choice secretary problem. Free order variants were studied extensively
for the secretary and prophet problems. Kesselheim, Kleinberg, and Niazadeh KKN
(STOC'15) initiated a study of randomness-efficient algorithms (with the
cheapest order in terms of used random bits) for the free order secretary
problems.
  We present an algorithm for free order multiple-choice secretary, which is
simultaneously optimal for the competitive ratio and used amount of randomness.
I.e., we construct a distribution on orders with optimal entropy
$\Theta(\log\log n)$ such that a deterministic multiple-threshold algorithm is
$1-O(\sqrt{\log k/k})$-competitive. This improves in three ways the previous
best construction by KKN, whose competitive ratio is $1 - O(1/k^{1/3}) - o(1)$.
Our competitive ratio is (near)optimal for the multiple-choice secretary
problem; it works for exponentially larger parameter $k$; and our algorithm is
a simple deterministic multiple-threshold algorithm, while that in KKN is
randomized. We also prove a corresponding lower bound on the entropy of optimal
solutions for the multiple-choice secretary problem, matching entropy of our
algorithm, where no such previous lower bound was known.
  We obtain our algorithmic results with a host of new techniques, and with
these techniques we also improve significantly the previous results of KKN
about constructing entropy-optimal distributions for the classic free order
secretary.
