Tournament Ranking with Expected Profit in Polynomial Time

An $O( n^3 \log n )$ algorithm is presented that, for a given tournament on n vertices, produces a ranking with fit at least $\frac{1}{2} \begin{pmatrix} n \\ 2 \end{pmatrix} + c_1 n^{3/ 2} $, where $c_1 = \frac{1}{8}\pi ^{ - 1 / 2} $.