A constructive lower bound on the quasi-Ramsey numbers and the tournament r
anking function was obtained in [S. Poljak, V. Rodl, and J. Spencer, SIAM J
. Discrete Math., (1) 1988, pp. 372-376]. We consider the weighted versions
of both problems. Our method yields a polynomial time heuristic with guara
nteed lower bound for the linear ordering problem.