Randomized metarounding

Let P be a linear relaxation of an integer polytope Z such that the integrality gap of P with respect to Z is at most r, as verified by a polytime heuristic A, which on any positive cost function c returns an integer solution (an extreme point of Z) whose cost is at most r times the optimal cost over P. Then for any point x* in P (a fractional solution), rx* dominates some convex combination of extreme points of Z. A constructive version of this theorem is presented here, with applications to approximation algorithms, and can be viewed as a generalization of randomized rounding. (C) 2002 Wiley Periodicals, Inc.*.