  This paper studies the problem of finding the exact ranking from noisy
comparisons. A comparison over a set of $m$ items produces a noisy outcome
about the most preferred item, and reveals some information about the ranking.
By repeatedly and adaptively choosing items to compare, we want to fully rank
the items with a certain confidence, and use as few comparisons as possible.
Different from most previous works, in this paper, we have three main
novelties: (i) compared to prior works, our upper bounds (algorithms) and lower
bounds on the sample complexity (aka number of comparisons) require the minimal
assumptions on the instances, and are not restricted to specific models; (ii)
we give lower bounds and upper bounds on instances with unequal noise levels;
and (iii) this paper aims at the exact ranking without knowledge on the
instances, while most of the previous works either focus on approximate
rankings or study exact ranking but require prior knowledge. We first derive
lower bounds for pairwise ranking (i.e., compare two items each time), and then
propose (nearly) optimal pairwise ranking algorithms. We further make
extensions to listwise ranking (i.e., comparing multiple items each time).
Numerical results also show our improvements against the state of the art.
